Finite element model for nonlinear analysis of steel–concrete composite beams using Timoshenko's beam theory Dinh Huynh Thai2 Bui Duc Vinh1 and Le Van Phuoc Nhan1 1 HCMUT, 268 Ly Thuong
Trang 1Finite element model for nonlinear analysis of steel–concrete composite
beams using Timoshenko's beam theory
Dinh Huynh Thai2) Bui Duc Vinh1) and Le Van Phuoc Nhan1)
1)
HCMUT, 268 Ly Thuong Kiet, Ho Chi Minh City, Viet Nam
2)
Hoang Vinh TRCC, 270A Tay Thanh, Ho Chi Minh City, Viet Nam
1) vinhbd@hcmut.edu.vn
ABSTRACT
This paper presents an analytical model for steel-concrete composite beams with partial shear interaction and shear deformability of the two components The model is obtained by coupling the Timoshenko’s beam for the concrete slab and steel girder (T-T model) The nonlinear material of concrete slab, steel girder and shear connectors are taken into account The stiffness matrix of the composite element with 16 DOFs is derived by the displacement based finite element formulation The numerical solutions are verified on simply supported and continuous beams The analytical results show good agreement with experimental data, they are also compared with the difference models
1 INTRODUCTION
Steel - concrete composite beams (CB) have been widely used in the construction industry due to the advantages of combining the two materials Modeling and analysis of steel–concrete composite structures have been proposed in the literature (Spacone and El-Tawil 2004)
Newmark et al (1951) analyzed CB with partial interaction The Newmark model couples two Euler–Bernoulli beams, i.e one for the reinforced concrete (RC) slab and one for the steel girder Since then, many researchers have been extended the Newmark’s model (Gattesco 1999, Dall’Asta and Zona 2002, Ranzi et al 2004) Recently, Ranzi and Zona (2007) introduced a beam model including the shear deformability of the steel component only This model was obtained by coupling an Euler–Bernoulli beam for the RC slab with a Timoshenko beam for the steel girder This parametric study was carried out using a locking-free finite element model under the assumption of linear elastic materials and considering the time-dependent behaviour
of the concrete Schnabl et al (2007) presents an analytical solution and a FE formulation for
CB with coupled Timoshenko beams for both components, the material models are limited on linear elastic behaviour The results showed that shear deformations are more important for high levels of shear connection degree, for short beams with small span-to-depth ratios, and for beams with high elastic and shear modules ratios
Trang 2
In this w
RC slab an
with partia
solution w
considered
element (F
continuous
2 ANALY
2.1 Mod
A typic
An ortho n
The comp
beam, refe
continuous
consists of
T-T model
2.2 Disp
The dis
shown in E
where v z(
displaceme
work, the pr
nd steel gird
al interaction
was reported
d for all com
FE) Four nu
s CB are pre
YTICAL M
del assumpti
cal steel–con
normal refere
osite cross s
erred to as A
s deformable
f the points
l can be foun
placement a
splacement f
Eq (1):
)
z represent
ents of the r
roposed mod der, it is refe
n, based on
d by Schna
mponents N umerical exa esented
MODEL
ions
ncrete CB wi ence system section is fo
As The com
e shear conn
in the YZ p
nd in work o
Fig.1 Typ
and strain fie
field of a g
( , )y z
⎧
⎪
⎪
= ⎨
⎪
⎪⎩
d d
d
ts the defle reference fib
del is formu ferred as (T–
kinematic a
abl et al (
Numerical so amples deal
ith prismatic
m {O; X, Y, Z ormed by the mposite actio nection at th
lane with y
of Schnabl et
ical compos
elds
eneric point ( , ) (
( ( , ) (
(
c
s
x
x
=
∀
=
∀
d d
ection of bo bers of the R
ulated by cou –T model) T assumptions
(2007) The olutions are ing with tw
c section is sh
Z} where i, j
e concrete sl
on between
he interface b
sc y
= andz∈
t al (2007)
site beam and
t P (x, y, z) ) [ ( )
) [ ( )
c c s s
j j
oth compone
RC slab and
upling Timo The governin
s substantiall
e nonlinear obtained by
o simply su
hown in Fig
j, k are the u
lab, referred the two com between the [0, ]L
d cross-secti
) of the CB
[0, ]
[0, ]
c c
s s
L
L
ϕ ϕ
∈
∈
ents; w z c( ) the steel gir
shenko beam
ng equation
ly similar to behaviour
y displaceme upported and
g 1 (Ranzi an
unit vectors o
d to as Ac, a mponents is two layers,
e main assum
ion
is defined b ( )]
( )]
z
z
k k
and w z s( ) rder, located
ms for both
s of CB mo
o the analyti
of material ent-based fin
d two two-sp
nd Zona 200
of axis X, Y, and by the st provided by whose dom mptions for
by vector d
are the ax
d at y and
the odel ical
l is nite pan
07) , Z teel
y a main the
d as
xial
y ,
(1)
Trang 3Translation
The dis
The sl
displaceme
where h c =
Based o
ly; ϕc( )z a
ns and rotati
splacement f
ip between
ents at their
( )z =s z( )
sc c
Fig
on the assum
nd ϕs( )z ar ions are sign field can be g
[ ( )
T z =
u
n the two interface, is ( ,
s y sc z
=
nd h s = y s−y
.2 Displacem med displacem
( , )
z y z
( , )
yz y z
re the rotat ned positive r grouped in th [w z c( ) w s( components given by ve )− dc(y sc, )z sc
y
ment field of ment field, t
z
z z
ε ε
⎧
⎪
⎪
⎪⎩
d k
=
j + k
tions of the respectively
he vector:
) v( )
s, which re
ector s:
)=[w ( )s z −w
f the T–T com the non-zero ( , ) ' ( , ) ( , ) ' ( , )
=
=
( , )
( , )
yz
yz
x y
x y
γ γ
⎧
⎪∀
⎪
= ⎨
⎪
⎪∀
⎩
k
top and bo
y as in Fig 2
( ) ( )
epresents th
w ( ) hc z − sϕs
mposite beam components
, [0, ]
, [0, ]
c
s
ϕ ϕ
∈
∈ ' , [0, ' , [0,
c
s
v
A z v
A z
ϕ ϕ
= +
= +
ottom layers
(Ranzi and
] )
he discontin
m model
s of the strai '
'
c
s
]
]
L
L
(
s, respective
Zona 2007)
nuity of ax )]
z k
in field are:
(4)
(5)
2)
(3)
ely
xial
Trang 4(6)
(7)
(8)
(9)
(10)
(6)
where εzc, εzsand γyzc,γyzs are the axial strains and the shear deformations of the two components, respectively
The strain field can be presented in the vector:
T
where εc( )z =w'cand εs( )z =w'sare the axial strains at the levels of the reference fibres of the two components respectively, θc( )z =ϕ'cand θs( )z =ϕ'sis the curvature of the RC slab and the steel girder
The vector of strain functions can be obtained from the vector of displacement functions by means of the relation:
ε = D u
where the matrix operator D is defined as:
1 1 0 h c h s
∂
∂
D
being ∂ the derivative with respect to z
2.3 Balance conditions
The principle of virtual work is utilized to obtain the weak form of the balance condition of the problem:
∂
∑ ∫ ∫ b d ∑ ∫ t d where b and t are the body and surface force respectively; (α =c s, )
From Eq (9) in weak form, the stress resultant entities, which are duals of the kinematic entities derived from the assumed displacement field, can be identified and grouped in the vector
r:
T
=
r
in which
Trang 5(11)
(12)
(14) (13)
(15)
(16)
(17)
z A
z A
yz A
α
α
α
σ σ τ
=
=
∫
∫
∫
Similarly, the external loads are written in the vector g:
T
g
in which
α
∂
∂
∂
Since Eq (9) can be rewritten in compact form as:
0L dz= 0L dz
∫ rDu ∫ gHu
with the matrix operator H defined as:
=
H
3 MATERIAL MODELS
3.1 Concrete
The stress-strain relationship suggested by the CEB-FIB Model Code (2010) is adopted in this paper for both compression and tension regions (Fig 3) The σc − relationship is εc approximated by the following functions:
• For εc < εc,lim :
2
c cm
k
η
= −⎜⎜ + − ⎟⎟
where: η ε ε= c / c1 and k =E ci/E c1
• For σct ≤0.9f ctm:
ct E ci ct
• For 0.9f <σ ≤ f :
Trang 6Fig 4
3.2 Stee
In the s
hardening
3.3 She
The con
(1971), is
where fmax
determined
Fig 3 St
4 a) Stress-s
el
study, the st
Fig 4 show
ear connecto
nstitutive re
given by:
x is the ultim
d from test
(a)
(a)
ct f
σ =
tress-strain d
train diagram
eel is model
ws the
stress-rs
elationship f
sc f
mate strength
)
1 0.1
0
cm
−
⎜
⎝
diagram for c
m for steel, b
led as an ela -strain diagr
for the stud
(
max 1
h of the stud
0.00015 0.00015 0.9−
concrete: a)
b) Load-slip
astic-perfect ram for steel
shear conne )
e−β δ α with
d shear conn
(b)
ct ctm ci
−
⎟
⎠
Compressio
diagram for
tly plastic m
in tension
ector was pr
u
δ δ≤ nector; and α
(b)
n, b) Tensio
r stud shear c
material incor
roposed by
α,β are co
(
(
on
connector
rporating str
Ollgaard et
efficients to
18)
19)
rain
al
be
Trang 74 FINITE
The po
same orde
problems:
(i) in
der
ecc
(ii) in
and
(Yu
(iii) in
pol
and
4.1 The
The sim
freedom (
considerat
rotations o
Based o
conditions
element ca
element is
10D
16D
The FE
in Fig 5b
the axial d
E ELEMEN
olynomial fu
er in each d
axial strain
rivative of t
centricity iss
the shear de
d the rotation
unhua 1998,
the interfac
lynomials of
d Zona 2004
e displaceme
mplest eleme
(DOF) Tha
tion Its shap
of componen
on the previo
s between th
an lead to p
discouraged
Table 1: D
DOF
DOF
E fulfilling th
which enhan
displacement
NT FORMU
unctions to a displacement (Eq 4), th the rotation sue (Gupta a
eformation (
n ϕ must be
, Mukherjee
ce slip (Eq
f the same o
4)
ent-based FE
ent (Fig 5a)
at is the a
pe functions nts (Table 1)
Fig.5 Fi ous consider
he different poor and un
d
Degrees of sh
c
w
1
2
he consistenc nces the ord
ts, rotations a
ULATION
approximate
t field, in fa
e first deriv
n ϕ must b
and Ma 1977
(Eq 5), the
e polynomial
and Prathap
3), the axia order in ord
E
) which can
t least requ are linear fu
inite element rations, this s displacemen nsatisfactory
hape function
s
w
1
2
cy condition der of the ap and to cubic
e displaceme act that need vative of the
be polynom
7, Erkmen an
e first deriva
ls of the sam
p 2001)
al displacem der to avoid
be derived uired DOFs unctions for
ts for the T–
simple 10 D
nt fields cou results Th
ns for the pr
v
1
3
ns of the disp proximated
c function for
ent are choo
d to avoid
e axial disp mials of the
nd Saleh 201
ative of the t
me order in o
ments w an
slip and cu
for the T–T
s for descr the axial dis
–T CB mode
OF FE does upled in the hus, the use
roposed T–T
c
ϕ
1
2 placement fi polynomials
r transverse
ose They ar the occurre
placement w
same order
12)
transverse d order to avoid
d the rotatio urvature lock
T model has ribing the p splacements,
l not satisfy t
e problem T
of the 10D
T finite eleme
ϕ 1 2 ield is the 16
s to paraboli displacemen
re must be ence of lock
w and the f
r to avoid
displacement
d shear lock
on ϕ must king (Dall’A
10 degrees-problem un ,deflection a
the consisten The use of t DOF T–T be
ents
s
ϕ
1
2 6DOF depic
ic functions
nt (Table 1)
the king first the
t v
king
be
Asta
-of-nder and
ncy this eam
cted for
Trang 8(20)
(21)
(22)
(23)
4.2 FE formulation
The displacement of the FE with a polynomial approximation of the displacement field is written as:
u = Nd
and relation of displacement and strain as in Eq (21):
= Nd=Bd
r = D DBd
D
ε
ε = the Virtual Work Principle Eq (9) becomes:
∫ DBd.Bd ∫ g.H N d
Since, the following balance equation is obtained:
e = e
K d f
where
0
L T
K B DB is stiffness matrix;
and
0L( )T
f H N g is the vector of the internal nodal forces
The calculation of load vector, internal nodal forces vector and stiffness matrix is performed
by means of numerical integration, using the trapezoidal rule through the thickness (the cross-section is subdivided into rectangular strips parallel to the x-axis) (Nguyen et al 2009) and by using the Gauss–Lobatto rule along the element length In computer code, five Gauss points are used in the 16DOF element The non-linear balance equation can be written in iterative form using the Newton–Raphson method
5 NUMERICAL EXAMPLES
The numerical solutions of the proposed model are compared against experimental data obtained by earlier experimental study In the fact that, a group of two CB which material limited in linear elastic range are investigated (Aribert et al 1983 and Ansourian 1981) Other group includes the simply-supported CB E1 tested by Chapman et al (1964) and the two spans
CB CBI tested by Teraszkiewicz (1967) are considered for nonlinear analysis
The proposed 16DOF beam element model is used to predict the elastic deflection of the simply supported composite beam tested by Aribert et al.(1983) The geometric characteristics and the material properties of the beam are shown in Fig 6 and Table 2
As shown in the figure a steel plate 120 × 8 mm is welded into bottom flange of the steel girder There are five rebars dia of 14mm, placing in at the mid-depth of the RC slab
Trang 9The be
proposed
experimen
placed at t
Distance
and the
Area
Second
Elastic m
Shear m
Shear bo
Fig 7 s
results of
model is
deformatio
compariso
shows both
Fig
eam is mod
model again
ntal data Be
the beam end
Table
Param
e between th
layer interfa
moment of a
modulus
modulus
ond stiffness
shows the lo
both model
closer to th
on of the c
on for the sli
h models pro
6 Geometric deled using nst the exis etween the s
ds
e 2: Mechani meter
he centroid o ace
area
s
oad–deflecti
ls are slight
he experime cross-section
ip distributio ovide almost
cal character six elemen sting EB-EB supports, fou
ical characte
of layer
c h c A c I c E c G
ion curve un tly more flex ental data th
n is taken in
on along the
t the identica
ristics of CB nts in order
B 8DOF mo
ur elements
eristics of CB
RC sl 50
82310
666.667
20000
8333
nder the poi xible than t han the EB-nto account
e beam leng
al slip distrib
(Aribert et
to compare odel (Dall’A
are used, an
B (Aribert et
lab
mm2
5
10 mm4 MPa
Pa
450
sc
int load It c the test data -EB 8DOF
t for each l gth at load l bution
al 1983)
e the perfo
Asta and Zo
nd two mor
t al 1983)
Steel 187
s
7220
s
1415
s
2000
s
8000
s
0 MPa
can be seen
a However model, bec layer Fig
level of 195
rmance of
ona 2002) a
re elements
l girder
mm
0 mm2
5
10
× mm4
000 MPa
00 MPa
that numeri
r, the propo ause the sh
8 presents
kN, the res
the and are
ical sed hear the sult
Trang 10Fig 9
those obta
stiffness (k
one evalua
related to
(loose con
of the effe
5.2 Two
The pro
CTB6, wh
considered
longitudin
the saggin
rebars are
(Nguyen e
Gs = 80.76
ig.7 Load–d
shows the m
ined with EB
ksc) The def
ated accordi
cases of low
nnection with
ct of shear f
Fi
o-span conti
oposed mode
hich was a
d The geom
nally reinforc
ng and hogg
25 mm and
et al 2011)
6 GPa; Ec = 3
eflection cur
mid-span de B-EB 8DOF flection pred ing to EB-E wer shear sti
h ksc = 1 MP flexibility of
ig.9 Mid-spa
inuous comp
el is now use part of the metric defin ced by rebar ging region
75 mm, res The materia
34 GPa; Gc =
rves
eflections ob
F model for d dicted by the
EB 8DOF m iffness, mon Pa) It can be
f the connect
an deflection
posite beam C
ed to simulat experiment ition of the
rs at the top The distanc pectively T
al parameter
= 14,167 GP
Fig
btained with different spa
e proposed m model, for an notonically re
e seen that p
ed members
n versus the s
te a two-spa tal program
e beam is il
p and bottom ces from th The shear bon
rs used in the
Pa
8 Slip distrib
h the propos an-to-depth r model is larg
ny value of t educe to the partial intera
s
span-to-dept
urian 1981)
an continuou carried out llustrated in
m with differ
e interface t
nd stiffness
e computer
bution along
sed model c ratios (L/H) ger than the the ratio L/H
e case of abs action results
th ratio
us steel–conc
t by Ansour Fig 10 T rent reinforc
to the botto
is assumed analysis are
g the beam
compared w and shear bo correspond
H The curv sent interact
s in a reduct
crete CB Be
rian (1981), The RC slab cement ratio
om and the
of 10,000 M
Es = 210 G
with ond ding ves, tion tion
eam , is
b is
o in top MPa Pa;