This study presentes a displacement-based finite element formulation for nonlinear
analysis of steel-concrete composite planar frames. A 6DOF super element for modelling
the composite beam is proposed. This element is able to allow for partial interaction,
material nonlinearity and semi-rigid connections. The nonlinear behaviour of materials of
this element derives entirely from the constitutive laws. This element also has the ability
to consider different behaviour of semi-rigid composite joints under sagging and hogging
moments. An algorithm for solving nonlinear analysis of composite structures with both
load control and displacement control is also proposed.
From the numerical examples, the validation of the formulation is demonstrated by
good agreement between the results from proposed formulation with experimental data
or results from other studies. The partial interaction, material nonlinearity and semirigid connections have significant influences on behaviour of composite beams as well as
composite frames, which are more sensitive to these factors in semi-rigid frame than in
rigid frame.
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Vietnam Journal of Mechanics, VAST, Vol. 33, No. 1 (2011), pp. 13 – 26
A 6DOF SUPER ELEMENT FOR NONLINEAR
ANALYSIS OF COMPOSITE FRAMES WITH PARTIAL
INTERACTION AND SEMI-RIGID CONNECTIONS
Le Luong Bao Nghi, Bui Cong Thanh
University of Technology, VNU-HCMC
Abstract. This paper presents a displacement - based finite element formulation for
nonlinear analysis of steel - concrete composite planar frames subjected to combined
action of gravity and lateral loads. A 6DOF super element is proposed for modelling
composite beam, allowing for partial interaction between the steel beam and the con-
crete slab, semi - rigid nature of beam to column composite connection and material
nonlinearity. The load control method and the displacement control method are utilized
for tracing the structural equilibrium paths, and the direct method is utilized for solving
the nonlinear problem. Numerical examples, concerning a two-span continuous composite
beam, a portal composite frame and a six - storey composite frame, are performed. The
results are compared with experience data or theoretical results from other studies and
are discussed for influences of the factors mentioned above on behaviour of composite
beams and composite frames.
Key words: Super finite element, composite constructions, partial interaction, semi -
rigid composite connections, nonlinear analysis.
1. INTRODUCTION
Steel - concrete composite constructions have been widely used in buildings as well
as bridges thanks to their ability to combine the advantages of both steel and concrete.
Therefore, the research into this type of construction to find out an efficient, robust and
accurate method for analysis of this type has always been necessary. The relative longitu-
dinal slip between the concrete slab and steel joist due to the deformability of the shear
connectors is referred to as the partial interaction (PI) in composite beams. This nature
has a significant influence on the behaviour of the composite beams and the composite
frames as well. Another nature of composite frame that affects highly the behaviours of
composite beams and frames is semi - rigid composite connections. It is very important to
account for the different behaviours of the composite connection under sagging moment
and hogging moment due to different contributions of the concrete and rebar to the bare
steel connection. In order to capture more precision in analysis of the composite frames,
especially in the limit states, it is very essential to account for nonlinear behaviour of steel,
concrete, shear connectors and semi - rigid connections. Two methods, which are usually
used for considering the material nonlinearity of member, are the lumped and distributed
14 Le Luong Bao Nghi, Bui Cong Thanh
models. The lumped models, which concentrate all material nonlinearity at the member
ends, seem to be appropriate for steel frame analysis. The distributed models, on the other
hand, are more accurate and rational due to monitoring entirely the material nonlinearity
along the member length by means of numerical integration. Thus, the distributed models
are efficient for modelling members made up of two different kinds of materials and for
members with cracked sections like composite members.
In this paper, the 8DOF displacement-based element proposed by Dall’Asta and
Zona [1, 2] is employed to modelling segments of planar composite beam allowing for PI.
The nonlinear behaviour of materials of this element derives entirely from the constitutive
laws by means of numerical integration. To improve the accuracy for the nonlinear analysis,
a number of the sequential 8DOF elements are combined to form the 8DOF super element
for modelling entirely one composite beam. To consider the semi-rigid nature of the com-
posite connection, firstly, two rotational springs are attached to the 8DOF super element’s
ends for modelling the bare steel connections. Then, additional rotational stiffness due to
the contribution of concrete and rebars is provided by the slab-to-column link. With that
description, the 8DOF super element is modified to formulate firstly a new 8DOF element
because of the attaching the springs, and then a 6DOF element by constrained conditions
obtained from slab-to-column links. Therefore, the 6DOF super element for modelling the
composite beam allowing for PI, material nonlinearity and semi-rigid composite connec-
tions is obtained. The direct method is utilized for solving nonlinear equilibrium equation,
and the load control method and the displacement control method are utilized for tracing
the structural equilibrium part.
2. FINITE ELEMENT FORMULATIONS
2.1. 8DOF element for modelling composite beam with PI
The 8DOF displacement based element [1, 2] (Fig. 1) is utilized for modelling a
segment of planar steel-concrete composite beam with partial interaction. The formulation
of this element is based on the Newmark kinematical model, where the Euler-Bernoulli
beam theory is used to model the two parts of the composite beam; the effects of the
deformable shear connection are accounted for by using an interface model with distributed
bond, preserving the contact between the components while allowing for the longitudinal
slip.
Fig. 1. 8DOF finite element
A 6dof super element for nonlinear analysis of composite frames... 15
The displacement formulation of the finite element method introduces a polynomial
approximation of the displacement field of the element:
u˜ = Nd (1)
(•˜ denotes the approximation of field ) where u is generalized vector representing the
displacement field:
uT =
[
w1 w2 v
]
(2)
(with v is the deflection of the beam and wα is the axial displacement of α−th component
(α = 1, 2) at its reference axis), d is the vector of the nodal displacements (Fig. 1) and N
is the matrix of shape functions. In this element, the deflection is described by the cubic
Hermite polynomial and the axial displacements of the two components are described by
linear functions.
Using the principal of virtual work, the equilibrium equation is written in the fol-
lowing form:
Ked = Pe (3)
where
Ke =
∫ Le
0
BTDBdz and Pe =
∫ Le
0
(H N)T pdz (4)
are the stiffness matrix and nodal force vector of the element. In above equations, B is
strain-displacement matrix defined from the following equation:
ε˜u = Du˜ = DNd = Bd (5)
where D is the differential operator representing compatibility conditions:
ε =
ε1
ε2
χ
s
=
∂ 0 0
0 ∂ 0
0 0 −∂2
1 −1 h∂
w1w2
v
= Du (6)
(with εα = w
′
α is the axial strain of α − th component (α = 1, 2) at its reference axis,
χ = −v′′ is the curvature, s is the slip and h is the distance of reference axes of the two
components). D is the matrix of constitutive relation between generalized stress r and
strain ε (r = Dε)
D =
EA1 0 ES1 0
0 EA2 ES2 0
ES1 ES2 EJ12 0
0 0 0 k
(7)
where
rT =
[
N1 N2 M12 q
]
(8)
and
EAα =
∫
Aα
EαdAα =
∫
Aα
dσzα
dεzα
dAα,
ESα =
∫
Aα
Eα (y − yα) dAα =
∫
Aα
dσzα
dεzα
(y − yα) dAα,
EJα =
∫
Aα
Eα (y − yα)
2
dAα =
∫
Aα
dσzα
dεzα
(y − yα)
2
dAα
(9)
16 Le Luong Bao Nghi, Bui Cong Thanh
(with EJ12 = EJ1 + EJ2, k is the connection stiffness, Nα is the axial forces on α − th
component (α = 1, 2), M12 is the summation of the bending moments Mα on α − th
component and q the interface shear force). H is the differential operator:
H =
1 0 0
0 1 0
0 0 1
0 0 ∂
(10)
and p is the external forces written in generalized vectors as follows:
pT =
[
pz1 pz2 py mx
]
(11)
where pz1, pz2, py, mx are two distributed loads, transverse distributed load and distributed
couple, respectively.
Fig. 2. Numerical integration for calculating stiffness matrix
To account the material nonlinearity of the element, the stiffness matrix and load
vector are formulated by means of numerical integration, using the trapezoidal rule through
the thickness (the cross-section is subdivided in rectangle strips parallel to the x-axis) and
the Gauss-Lobatto rule along the element length as shown in Fig. 2.
2.2. Super element for modelling composite beam with PI and semi-rigid
connections
Due to using the low order shape functions for the 8DOF element, it is necessary to
use a number of the 8DOF elements for modelling one composite beam in order to describe
the displacement field better as well as increase the accuracy in nonlinear analysis. That
means the composite beam is subdivided along its length into a number of segments, each
of which is modelled by one 8DOF element (Fig. 3a). However, this subdivision brings
about the difficulties in numerical computation due to the large size of the global stiffness
matrix, and in managing the input and output data. To overcome these problems, these
discrete 8DOF elements are combined to form one element referred to as 8DOF super
element (Fig. 3b) for entirely modelling the composite beam. The stiffness equation for
the discrete 8DOF elements is given by:
K · d =
[
K11 K12
K21 K22
]{
d1
d2
}
=
{
P1
P2
}
(12)
where K is the overall stiffness matrix obtained by assembling the stiffness matrices of
internal elements, d1 and P1 are the displacement and force nodal vectors of the nodes
A 6dof super element for nonlinear analysis of composite frames... 17
at member’s ends, d2 and P2 are the displacement and force nodal vectors of the internal
nodes. Through static condensation, the stiffness matrix and the force nodal vector (not
including reaction force vector) of the super element are defined:
Ke(8x8) = K11 −K12K
−1
22
K21 and P
′
e(8x1) = P
′
1−K12K
−1
22
P2 (13)
Fig. 3. The 8DOF super element for composite beam
In this study, a model of composite joint is proposed as shown in Fig. 4. In this model,
the bare steel connection is modelled by a rotational spring characterized by moment-
rotation curve, and the contribution of the reinforced concrete slab to rotational stiffness
is considered by a slab-to-column link. This link is the assumption that the node associated
with the reinforced concrete slab is connected to the point, at which the reference axes
of the column and of the slab intersect as shown in Fig.4. This model makes it possible
to consider the different behaviours of the composite joint under sagging and hogging
moments due to different behaviours of concrete in compression and tension.
Fig. 4. Model of semi-rigid composite joint
To account the semi-rigid nature of composite joint in composite beam, firstly, two
rotation springs are attached at the ends of the 8DOF super element for modelling the
bare steel connection (Fig. 5a). The stiffness equation for the composite beam element
and the two springs shown as three separate elements in Fig.5a are given by:
Ke(8x8)
R1 −R1
−R1 R1
R2 −R2
−R2 R2
d(8x1)
d9
d10
d11
d12
=
Pe(8x1)
P9
P10
P11
P12
or K(12x12) · d(12x1)= P(12x1)
(14)
18 Le Luong Bao Nghi, Bui Cong Thanh
where R1 and R2 are the stiffness of the two springs. Then the three elements are connected
together to form one element having 10 degrees of freedom (DOFs) as shown in Fig. 5b.
The relationship between the 12 DOFs of the element in Fig. 5a and the 10 DOFs of the
element in Fig.5b is defined by the transformation matrix:
T(12x10) =
1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0 0
(15)
Therefore, from Eqs. (14) and (15), the stiffness matrix and the force nodal vector
of the element in Fig. 5b is given by:
K(10x10) =
(
T(12x10)
)T
·K(12x12) ·T(12x10) and P(10x1) =
(
T(12x10)
)T
·P(12x1) (16)
To obtain the 8DOF element as shown in Fig. 5c, static condensation is employed
to eliminate the DOFs 9’ and 10’of the element in Fig. 5b. Thus, the stiffness equation for
the element in Fig.5b is expressed as:[
K(8x8) K(8x2)
K(2x8) K(2x2)
]{
d(8x1)
d(2x1)
}
=
{
P(8x1)
P(2x1)
}
(17)
where d(8x1) and P(8x1) are the displacement and forces corresponding to the exterior
Fig. 5. The 8DOF super element for composite beam with semi-rigid connections
DOFs 1’ to 6’, and d(2x1) and P(2x1)are the displacements and forces corresponding to the
interior DOFs 7’ to 8’. Therefore, from Eqs. (17), the stiffness matrix and the force nodal
A 6dof super element for nonlinear analysis of composite frames... 19
vector of the 8DOF element in Fig. 5c is given by:
KSRe(8x8) = K(8x8) −K(8x2) ·K
−1
(2x2) ·K(2x8) and P
SR
e(8x1)= P
′
(8x8)−K(8x2)K
−1
(2x2)P(2x8) (18)
b
Fig. 6. The 6DOF super element for composite beam with semi-rigid connections
To consider the contribution of the reinforced concrete component to the composite
joint, the slab-to-column links proposed above are employed. On the column’s reference
axes, if the differences in deflection due to bending moment within the distances h (Fig.
6a) are inconsiderable, the links lead to constrain equations of the DOFs of the element
in Fig. 6a:
d1 = d2 − d4 · h and d5 = d6 − d8 · h (19)
Thanks to these constrain equation, the 6DOF element in Fig.6b is obtained by
eliminating the DOF’s 1 and 5 in the 8DOF element in Fig. 6a, and have the stiffness
matrix and the force nodal vector given by:
Ke(6x6) =
(
T(8x6)
)T
·Ke(8x8) ·T(8x6) and P(6x1) =
(
T(8x6)
)T
·P(8x1) (20)
where T(8x6) is the transformation matrix representing the relationship between the 8
DOFs of the element in Fig. 6a and the 6 DOFs of the element in Fig.6b:
T(8x6) =
1 0 −h 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 −h
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
(21)
Now, the 6DOF super element for modelling composite beam with PI, material non-
linearity and semi-rigid composite joints has been formulated. The transforming the 8DOF
element in Fig. 6a into the 6DOF element in Fig. 6b also make possible the assembling
the DOFs of the beam element’s nodes to those of the column’s nodes which only have 3
DOFs.
20 Le Luong Bao Nghi, Bui Cong Thanh
3. NONLINEAR ANALYSIS OF COMPOSITE FRAMES
An important aim of the nonlinear analysis is to trace the structural equilibrium
path until failure is reached. Herein, the load control method and displacement control
methods are implemented for this aim. In the load control method (Fig. 7a), the displace-
ment nodal vector corresponding with a fixed load level is solved through an iterative
procedure. This method is not suitable for describing equilibrium paths with weak load
l 1
l 2
l n
1
q
u
2
q
u
q
n
u
L
o
ad
le
v
el
Limitpoint
Equilibrium path
Displacement
(b) Displacement control
l 1
l 2
l n
L
o
ad
le
v
el
Equilibrium
path
Limit point
Displacementu1 u2
(a) Load control
Fig. 7. Methods for tracing the equilibrium path
Fig. 8. The direct method
level variations and cannot trace softening equilibrium paths. In the displacement control
method (Fig. 7b), a chosen component of the displacement nodal vector is fixed, and the
load level and the other displacement components is solved through iterative procedure
to correspond with the fixed displacement component. This method can trace equilibrium
paths with weak load level variations and softening equilibrium paths as well. The iterative
procedure for solving nonlinear equilibrium equation is the "direct method" (Fig. 8).
4. NUMERICAL EXAMPLES
To demonstrate the robustness and validity of the proposed method, a computer
program, called COMFAD, is developed and employed for nonlinear analysis of a number of
composite structures. The results by the proposed method are compared with experimental
as well as numerical results by other authors, and are discussed for influences of PI, semi-
rigid joints and material nonlinearity on behaviour of composite beams and composite
frames.
A 6dof super element for nonlinear analysis of composite frames... 21
d
te
Fig. 9. Nonlinear constitutive laws
Table 1. Geometrical properties of the test beam
Span length (mm) 4500
Concrete slab Thickness (mm) 100
Width (mm) 800
Steel beam Section (mm) HEA 200
Reinforcement Top (mm2) 804
Bottom (mm2) 767
Shear stud Kind 19x75
Spacing (mm) 350
Number 84
The constitutive laws adopted to describe nonlinear behaviours of materials are
presented as follows. Elastic-perfect plastic-hardening constitutive law cited by [1] is used
for steel and rebars (Fig. 9a). The nonlinear law suggested by CEB-FIP Model Code 1990
[5] is adopted for concrete under compression while null strength is used under tension,
(Fig. 9b). The Ollgaard constitutive law cited [1] is employed for the shear connection (Fig.
9c). The rotational stiffness of the spring for modelling the bare steel joint is characterized
by moment-rotation curve suggested by Eurocode 3 [6] (Fig. 9d).
4.1. Example 1: Two-span continuous composite beam
The composite beam CTB4, tested by Ansourian [7], was simulated by the method
in this study (Fig. 10). The geometrical and material properties are reported in Table 1
and Table 2.
Fig. 11 shows the load versus midspan deflection plots by proposed method, general
method by Ranzi [8] and experimental data by Ansourian. It can be seen that there are
22 Le Luong Bao Nghi, Bui Cong Thanh
Table 2. Material properties of the test beam
Concrete Compressive strength fc (MPa) 34.0
Steel Yield stress fν (MPa) Flange 236
Web 238
Reinforcement 430
Ultimate tensile stress fu (MPa) Flange 393
Web 401
Reinforcement 533
Elastic modulus Es (MPa) 206
000
Shear stud Qu (kN) 110
β (mm−1) 1.2
α 0.85
Fig. 10. The two-span continuous beam
good agreements between the results of proposed method and Ansourian’s experiment
in case of material nonlinearity and between the results of proposed method and Ranzi’s
method in case of linear material assumption. It also shows that there is obvious difference
between two cases of material linearity and nonlinearity when the load W is above 200
kN.
Midspandeflection (mm)
0 10 20 30 40 50 60
0
50
100
150
200
250
300
Proposed, non-linear
Proposed, linear
Ranzi, linear
Ansourian experiment
L
oa
d
W
(k
N
)
0 10 20 30 40 50 60
0
50
100
150
200
250
300
350
Khong nut, nsc=10(neo/1m)
Nut, nsc=10(neo/1m)
Khong nut, nsc=5(neo/1m)
Nut, nsc=5(neo/1m)
Uncracked c 10 (studs/m)
Cracked, nsc = 10 (studs/m)
Uncracked, c (studs/ )
Cracked, nsc =5 (studs/m)
Midspan deflection (mm)
L
oa
d
W
(k
N
)
Fig. 11. Comparision between numerical
results and Ansourian’s experimental results
Fig. 12. Midspan deflection with cracked
and uncracked concrete
A 6dof super element for nonlinear analysis of composite frames... 23
Fig. 12 shows the load versus midspan deflection plot with the assumption of cracked
and uncracked concrete sections and the different degree of shear interaction. It can be
seen in the figure that the crack and uncrack assumption and the shear interaction has
significant influences on the behaviour of beam deflection as well as the ultimate load.
0 1 2 3 4 5 6 7 8 9
-1.5
-1
-0.5
0
0.5
1
1.5
Khongnut,nsc=10(chot/1m)
Nut,nsc=10(chot/1m)
Khong nut,nsc=5(chot/1m)
Nut,nsc=5(chot/1m)
Uncracked, 10 (studs/m)
Cracked, nsc = 10 (studs/m)
Uncracked, (studs/ )
Cracked, ns =5 (studs/m)
S
li
p
(m
m
)
x (m)
Fig. 13. Slip with cracked and uncracked
concrete
Fig. 14. Portal composite frame
Fig. 13 shows the variation of longitudinal slip along member length. The slip is
also considerably affected by the cracked and uncracked assumption as well as the shear
interaction. It can also be seen that the higher the degree of interaction is, the less sensitive
to the consideration for craking the slip is.
4.2. Example 2: Portal composite frame
Fig. 14 shows the portal composite frame that was analyzed by Liew and Chen
(2001) [9]. That consists of a composite beam rigidly connected to two steel columns. The
geometrical of the members are shown in Fig. 14. The strength of concrete fc = 16 MPa,
the yield strength of steel fy = 252.4 MPa, and the elastic modulus of steel E = 2x10
5
MPa. The degree of shear connection is N/Nf = 0.9 where Nf is the stud connections
designed to obtain full interaction according to EC4.
The load versus lateral displacement curves in Fig. 15 indicates that good agreement
between the results by the proposed formulation in this study and those by Liew and Chen.
It can be seen that the pure steel frame collapses at P = 60.2 kN while the composite
frame at P = 82.7 kN. The limit load of the frame, thanks to the composite action, is
increased by 37%. However, this composite action is effective to increase the limit load
when slab-to-column links are considered. If these links are not considered, the limit loads
of the pure steel frame and of the composite frame are nearly the same as shown in Fig.
15.
The distributed load on the composite beam versus the midspan deflection with dif-
ferent degree of interaction is plotted in Fig. 16. The results indicate that shear interaction
has significant influence on the deflection of the composite beam subject to distributed
load as well as on the limit load.
24 Le Luong Bao Nghi, Bui Cong Thanh
Fig. 15. Load - Displacement curves Fig. 16. Mid-span deflection of composite
beam
4.3. Example 3: Six-storey composite frame
Analyzed herein is a six-storey composite frame with geometrical properties shown
in Fig. 17, which has been by a number of researchers as benchmark examples [10, 11].
The strength of concrete fc = 25 MPa, the yield strength of steel fy = 235 MPa, and the
6
x
3
.7
5
=
2
2
.5
m
IPE240
IPE300
IPE300
IPE360
IPE360
IPE400
31.7kN/m
49.1kN/m
49.1kN/m
49.1kN/m
49.1kN/m
49.1kN/m
10.23 kN
20.44 kN
20.44 kN
20.44 kN
20.44 kN
20.44 kN
HEB160
HEB160
HEB220
HEB220
HEB220
HEB220 HEB260
HEB260
HEB240
HEB200
HEB240
HEB200
2x6 =12m
D1 D2
D3 D4
D5 D6
D7 D8
D9 D10
D11 D12
6
x
3
.7
5
=
2
2
.5
m
1100
120
4d8 + 4d12
36.169877D11, D12
47.7815375D7, D8,
D9, D10
59.7424582D3, D4,
D5, D6
67.7930192D1, D2
Steel-steel
moment
resistance
Mu (kNm)
Initia l stiffness
Ri,i ni (kNm/rad)
Beam
Table 3. Characteristics ofsteel connections
Fig. 17. Six-storey composite frame
elastic modulus of steel E = 2x105 MPa. The shear connections have: dsc= 12.7mm, Qu =
66 kN; β=0.8 mm-1, α= 0.4. The initial stiffness and the ultimate of bare steel connection
are calculated by COMFAD according to EC3 and are shown in Table 3.
Fig. 18 shows the curves of load level versus lateral displacement at the top of frame
with different degree of shear interaction between steel beam and concrete slap. The case
of the steel frame with no concrete slab is also included. It can be seen that the composite
action of composite beam play an important role in the stiffness of the frame and in the
ultimate load as well. In case of semi-rigid frame, the figure shows that the two curves
from steel frame and no-interaction frame nearly coincide. The results also show that the
A 6dof super element for nonlinear analysis of composite frames... 25
displacement is less sensitive to the degree of shear interaction in rigid frame than in
semi-rigid frame.
0 20 40 60 80 100 120 140 160 180
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Chuyen vi ngang dinh khung (mm)
H
e
s
o
ta
i
tr
o
n
g
Khung thep
Khong tuong tac
nsc=5(neo/1m)
nsc=10 (neo/1m)
0 20 40 6 0 80 100 120 140 160 180 200 220 240
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Chu yen vi ngang d inh khu ng (mm)
H
e
s
o
ta
i
tr
o
n
g
Khung thep
Khong tuong tac
nsc=5(neo/1m)
nsc=10 (neo/1m)
Figure 18. Top later deflection vs load level
a) Rigid frame b) Semi-rigid frame
steel frame
no interacti
nsc= 5 (studs/ )
nsc= 10 (studs/ )
steel frame
no interaction
nsc=5 (studs/ )
nsc=10 (studs/m)
L
o
a
d
le
v
e
l
Lateral top deflection (mm)
L
o
a
d
le
v
e
l
Lateral top deflection (mm)
Fig. 18. Top lateral displacement vs load level
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Rigidframe,uncracked
Rigid frame,cracked
Semi-rigid frame,uncracked
Semi-rigid frame,cracked
Rigid frame,linear material
Semi-rigid frame,linear meterial
Lateral top deflection (mm)
L
o
a
d
le
v
el
M
o
m
en
t
(k
N
m
)
Fig. 19. Influences of cracked and uncracked
sections on lateral displacement
Fig. 20. Connection stiffness under hogging
and sagging moment
Fig. 19 shows the load level versus later top deflection plot in cases of rigid and
semi-rigid connections with the assumptions of material linearity and nonlinearity with
cracked and uncracked concrete sections. It can be seen in the figure that the crack and
uncrack assumptions plots obviously different results of lateral deflection and ultimate
load as well. The results also show such differences are more distinct in rigid frame than
in semi-rigid frame.
Fig. 20 shows the behaviour of the composite connections under sagging and hogging
moment. It can be seen that the stiffness of the composite joint is visibly larger than the
stiffness of bare steel connection’s contribution. The results also indicate the considerably
different behaviour of the composite joints under sagging and hogging moments.
5. CONCLUSIONS
This study presentes a displacement-based finite element formulation for nonlinear
analysis of steel-concrete composite planar frames. A 6DOF super element for modelling
the composite beam is proposed. This element is able to allow for partial interaction,
material nonlinearity and semi-rigid connections. The nonlinear behaviour of materials of
this element derives entirely from the constitutive laws. This element also has the ability
to consider different behaviour of semi-rigid composite joints under sagging and hogging
26 Le Luong Bao Nghi, Bui Cong Thanh
moments. An algorithm for solving nonlinear analysis of composite structures with both
load control and displacement control is also proposed.
From the numerical examples, the validation of the formulation is demonstrated by
good agreement between the results from proposed formulation with experimental data
or results from other studies. The partial interaction, material nonlinearity and semi-
rigid connections have significant influences on behaviour of composite beams as well as
composite frames, which are more sensitive to these factors in semi-rigid frame than in
rigid frame.
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Received January 20, 2010
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