Long-Term Behavior of Timber-Concrete Composite Beams.
I: Finite Element Modeling and Validation
Massimo Fragiacomo1; and Ario Ceccotti2
Abstract: The first part of two companion papers deals with the numerical modeling of
Timber-Concrete Composite beams (TCC’s) under long-term loading. All phenomena
affecting the long-term behavior of timber, concrete and the connection system, such as
creep, mechano-sorptive creep, shrinkage/swelling and temperature variations, are fully
considered. The structural problem is solved through a uniaxial finite element model
with flexible connection and a step-by-step numerical procedure over time. The
important role played by the environmental thermo-hygrometric variations on TCC’s is
highlighted through some analyses. The proposed numerical procedure is validated on
two long-term experimental tests in outdoor conditions. Despite some uncertainties in
environmental conditions and material properties, a good fit between experimental and
numerical results is obtained. A parametric analysis is performed in the second part,
showing the contribution of different rheological phenomena and thermo-hygrometric
variations on beam deflection and connection slip. Based on results carried out, a
simplified approach for long-term evaluation of TCC’s is then proposed.
CE Database keywords: Composite beams; Concrete; Creep; Finite element method;
Time dependence; Shrinkage; Timber construction; Wood.
1
Research Engineer, Department of Civil Engineering, University of Trieste, Piazzale Europa 1, 34127
Trieste, Italy, e-mail fragiacomo@dic.univ.trieste.it, tel. +39 040 558 3845, fax +39 040 54413.
2
Director, IVALSA Trees and Timber Institute, National Research Council of Italy, via Madonna del
Piano, 50019 Sesto Fiorentino (Florence), Italy, e-mail ceccotti@ivalsa.cnr.it, tel. +39 0461 660111, fax
+39 0461 650045.
1
Introduction
The Timber-Concrete Composite beam (TCC) represents a construction technique
widely used in both new and existing constructions for upgrading strength and stiffness.
This technique consists of connecting an existing or new timber beam with a concrete
slab cast above a timber decking by means of a connection system. A steel mesh is
generally placed into the concrete flange in order to resist possible tensile stresses due to
slab bending and to reduce the crack width. Several types of connection systems are
available to link the timber beam with the concrete flange (Ballerini et al. 2002, Balogh
et al. 2002). They generally cannot prevent a relative slip between the two linked parts,
thus the connection should be regarded as flexible when studying the TCC.
According to design codes (C.E.N. 1995, 1996), both serviceability and ultimate
limit states have to be checked. For medium to long span beams and/or heavy
environmental conditions (e.g. bridges or roof structures), the serviceability limit state
of maximum deflection may be the most severe design criterion. Thus it is important to
investigate the behavior of such structures under long-term loading. All the materials
employed to construct TCC’s, i.e. concrete, timber and the connection system,
demonstrate important time-dependent phenomena, which affect both strain and stress
distribution. The creep and shrinkage of concrete are well-known phenomena, as is the
creep of timber. The change of environmental relative humidity also affects the timber
behavior, since it increases the delayed strains under constant load (the so-called
mechano-sorptive effect), causes shrinkage/swelling and influences the Young modulus
(Ranta Maunus 1975, Mårtensson 1992, Toratti 1992, Hanhijärvi and Hunt 1998). The
creep and mechano-sorptive phenomena also occur in the connection system, as was
2
demonstrated by some recent experimental tests (Bonamini et al. 1990, Kenel and
Meierhofer 1998, Amadio et al. 2001).
The structural problem is, therefore, rather complex and a numerical procedure
has to be employed in order to find accurate solutions. Few numerical approaches have
been proposed so far. Capretti (1992) schematized the TCC as a Vierendeel beam, using
the Ranta Maunus model (1975) in order to describe the rheological behavior of timber
and connection. Kuhlmann and Schänzlin (2001) developed an algorithm based on the
finite difference method that considers all the aforementioned time-dependent
phenomena and employs the Hanhijärvi and Hunt model (1998) for describing the
mechano-sorptive creep of timber. Said et al. (2002) performed a three-dimensional
numerical analysis of TCC’s using the Abaqus explicit finite element code.
The purpose of the first part of two companion papers is to validate and calibrate
a numerical procedure for long-term analyses of TCC’s. The procedure is based on a
uniaxial finite element with smeared connection. The FE model is general, however
adopted assumptions and results apply for simply supported TCC’s, the most important
and common static scheme. All time-dependent phenomena of the component materials,
such as creep, mechano-sorptive creep and shrinkage/swelling, are fully considered
through a numerical algorithm based on a step-by-step procedure over time. The
influence of environmental thermo-hygrometric variations is carefully evaluated.
Experimental and numerical results are critically compared in order to validate the
proposed model and to highlight possible shortcomings. The model is used in the
second part to perform a parametric analysis showing the contribution of each
rheological phenomena and thermo-hygrometric variations. Based on results carried out,
a simplified approach for long-term analyses of TCC’s is then proposed.
3
Finite element model
The finite element used to model the TCC is displayed in Fig. 1. It is constituted by a
lower timber beam linked to an upper concrete flange by means of a continuous spring
system. Such a spring system represents the connection by hypothesizing the connectors
as smeared along the beam axis. Two layers of reinforcement may be placed inside the
concrete slab. The timber and concrete cross-sections are divided into horizontal and
vertical fibers in order to consider different properties along the height and the width.
Kinematic hypotheses
The kinematic hypotheses, similar to those adopted by Newmark et al. (1951) for steelconcrete composite beams, are:
-
negligibility of shear strains for both timber beam and concrete flange;
-
equal vertical displacements for timber and concrete, i.e. absence of uplifting:
vc  v w  v (Fig. 1);
-
preservation of the plane cross-sections for both timber and concrete flange;
-
no slip between reinforcement and concrete.
Let now Gc, Gw be, respectively, the geometrical centers of the concrete area without
reinforcement and timber cross-section, with H the distance between them (Fig. 1).
According to the aforementioned hypotheses, the strain-displacement laws for a generic
point P x, y, z  in the concrete flange (subscript c ) and timber beam (sub. w ) are:
 c  u c  y c v 
(1)
 w  u w  y w v
(2)
4
where  , y represent the strain and the distance of the point P from G, u , v denote the
axial and vertical displacement of G, and x is the abscissa of the cross-section. For the
connection, the relative slip between concrete flange and timber beam s f is given by:
s f  Hv   u c  u w 
(3)
Constitutive law for concrete
It is well known that concrete shows important time-dependent phenomena, such as
creep and shrinkage. In addition, the cracking event should be considered if the material
is subjected to large tensile stresses. For simply supported TCC’s the slab is subjected to
bending and compression. Thus the tensile stresses due to long-term loading, if existing,
are generally lower than the tensile strength of concrete. The cracking phenomenon has,
therefore, been neglected while concrete is assumed to be a viscoelastic material with
inelastic strains due to shrinkage and thermal variations. This implies that the proposed
model can be used for simply supported TCC’s, which is the most common and
important case. Let Rc ,  c ,  c s ,  c T , t 0 , t ,  be the relaxation function, the total
strain, the shrinkage strain, the inelastic strain due to thermal variations, the initial time,
the final time and the current time respectively. The constitutive equation for concrete is
the integral-type relaxation law:
 c t    Rc (t , )d  c ()   c s ()   c T ()
t
t0
(4)
with
5

 c T     c T dTc '
t0
(5)
where  c T is the thermal expansion coefficient and Tc  ' is the temperature of concrete
at the time  ' . In order to solve this equation through an effective computational
approach, the relaxation function is expressed as a sum of exponential functions:
N
Rc t ,    E c n   e
 t  


  
 cn 
(6)
n 1
that is the same as to use the generalized Maxwell model to represent the rheological
behavior of concrete. The parameters E c n () and  c n , i.e. the Young modulus at the
instant  after the concrete casting and the relaxation time of the n th Maxwell chain,
were evaluated by Lacidogna (1994) for the CEB-FIP M.C. 90 creep prediction model
(C.E.B 1993). They are expressed as a function of the average environmental relative
humidity RH , the medium cylindrical compressive strength f cm and the notational size
of member h  2 Ac / u , where Ac and u are the area and perimeter of the cross-section
exposed to the atmosphere, respectively.
Constitutive law for timber
The rheological behavior of timber is rather complex because it is influenced by the
moisture content u , which is the ratio between the mass of water content and the mass
of dried timber. The quantity u varies in time and over the cross-section Aw according
to the diffusion laws (Toratti 1992):
6
u   u    u 
  D    D  P( y, z )  Aw
t y  y  z  z 
(7)
qu  S u eq  u  P( y, z )  Aw
(8)
u y, z, t 0   u 0  y, z  P( y, z )  Aw  Aw
(9)
where D , S are the diffusion coefficient and the surface emissivity, parameters that
depend on the properties of timber, q u is the moisture content flux through the
boundary of the cross-section Aw , u0 is the moisture content distribution at the initial
time of analysis t 0 , and u eq is the timber moisture content in equilibrium with the
atmosphere, given by:
u eq 
0.01 RH
(0.00084823 RH  0.11665 RH  0.38522)
2
(10)
where RH is the environmental relative humidity (in percentage). In these formulae,
the influence of the temperature T on u has been disregarded, being less important
compared to the influence of RH (Toratti 1992). Equations (7) to (9) have been solved
by dividing the cross-section in cells and by using an explicit method of integration in
time with a stability criterion (Fragiacomo 2000), by which the quantity u  u y, z, t 
can be evaluated once the history RH  RH t  is known.
Several rheological models were proposed for timber, both linear (Ranta
Maunus 1975, Mårtensson (1992), Toratti 1992) and non-linear (Hanhijärvi and Hunt
1998). In this paper the Toratti model has been employed. This model is linear with
respect to the stress and therefore can be considered as a hydro-viscoelastic model. It
was demonstrated to provide a good agreement with experimental bending tests, as long
7
as the maximum stress is less than 20 % of the timber strength. This condition is
generally satisfied when studying TCC’s subjected to the service load. Let now  w ,  w
be the total strain and stress at the instant t , and J w t , , u  the creep function of timber,
given by:
t  
1
1

J w t , , u   J w 0 (u )  J w c (t , ) 

 
E w0 1  k u u  E w0 1  k u u ref   t d 
m
(11)
with E w0 the Young modulus of dried timber, u ref , k u , t d and m material parameters
evaluated by Toratti (1992). The constitutive equation can be written in integral form as:
t
t
t
t0
t0
 w t    J w 0 u  d w    J w c t ,  d w     w  dJ w 0 u  
t0

 J  1  e
t0


w
t
 c
 w
t

du  1  

t
t
t

d w   t bw  w du   t  w u du   t  w T dTw 
0
0
0

(12)
In this equation:
-
the sum of the first two integrals represents the viscoelastic strain;
-
the third integral accounts for the dependence of the Young modulus on the
moisture content;
-
the fourth integral represents the mechano-sorptive term, where J w and c w are
material parameters;
-
the fifth integral takes into account the dependence of shrinkage/swelling on the
total strain  w , where bw is a material parameter;
8
-
the sixth and seventh integrals represent the inelastic strains due to
shrinkage/swelling and thermal expansion, where  w u ,  w T , Tw   are the moisture
and thermal expansion coefficients, and the temperature of timber at the instant 
respectively.
In order to make the computational process more effective and to reduce the
computer used memory, the timber creep function is expressed as a sum of exponential
functions, which is equivalent to the use of a generalized Kelvin model:
 t  



  

J w (t , , u )  J w 0 u   J w 0 u ref  J w n 1  e  wn  


n 1


M
(13)
where the parameters J w n and  w n were evaluated for every n th Kelvin element by
Toratti (1992) for the power-type hereditary creep function, generally employed for
timber, given by Eq. (11).
Constitutive law for connection and reinforcement
Unlike concrete and timber, for which the rheological behavior was deeply investigated,
few experimental tests under long-term loading have been performed so far on the
connection systems (Bonamini et al. 1990, Kenel and Meierhofer 1998, Amadio et al.
2001). These tests have demonstrated that connection creeps, even more than timber,
and the viscous behavior is influenced by timber moisture content changes. No hydroviscoelastic model has been proposed, however studies are in progress (Amadio et al.
2001). In this paper, the Toratti rheological model has been employed to describe the
creep and mechano-sorptive creep of connection system since the connection stiffness is
9
mainly affected by the deformability of timber. The non-linear behavior under shortterm load has been disregarded. This assumption is reasonable when the connector shear
force is quite low, such as that produced by the service load.
The creep function is given by:
J f (t , )  J f 0
1
1
 J f c t ,  
 ck
Kf
Kf
 t  



 f 

n 

J f n 1 e



n 1


M
(14)
where ck represents a possible creep amplification factor with respect to the timber, the
parameters J f n and  f n are evaluated on the basis of long-term push-out tests or, in
absence of experimental data, are assumed equal to those employed for timber, and
K f  k f / i f is the equivalent smeared shear stiffness, with k f connector shear stiffness
and i f connector spacing along the beam axis. The constitutive law in integral form is:
 c


 f  du  1   
s f t    J f 0 dS f    J f c t ,  dS f   J  1  e  
dS f  
t0
t0
t0


t
t
t

f
t
(15)
t
  b f s f  du 
t0
where S f is the shear force per unit length carried by the connection system, s f is the
relative slip between concrete flange and timber beam, J f , b f and c f are parameters
equal to the ones assumed for timber.
The reinforcement is considered as a linear-elastic material with possible
inelastic strains due to thermal variations.
10
The solving linear system
The integral equations (4), (12) and (15), which represent the constitutive laws for
concrete, timber and connection, are transformed in algebraic equations by dividing the
whole reference period t 0 , t  in time steps t k  t k  t k 1 , with k  1,2,..., m and by
applying the trapezoidal rule for every step k . The strain-displacement laws (Eqs. (1) to
(3)) written in incremental form are then substituted into those equations. The axial
displacement increments of the geometrical centers Gc and Gw, and the vertical
displacement increment of a cross-section at abscissa x are given by:
ui k x   N i x u i k
i  c, w
vk x   N v x v k
(16)
(17)
where Nx  are the shape function matrixes and u i k , v k are the nodal displacement
increment vectors. The assumed shape functions are (Amadio and Fragiacomo 1993)
quadratic for axial displacements ( N i ) and cubic for vertical displacements ( N v ). By
substituting Eqs. (16) and (17) into Eqs. (1) to (3) written in incremental form and by
applying the Principle of Virtual Work it is hence possible to draw the solving linear
system for the step k (Fragiacomo 2000).
Effect of environmental thermo-hygrometric variations
Before validating the numerical procedure on experimental tests, it is useful to discuss
on the effects produced by environmental thermo-hygrometric variations on TCC’s. The
different physical properties of timber and concrete concerning the heat and moisture
11
diffusion processes lead to diverse responses of these materials with the environmental
thermo-hygrometric variations being the same. Besides, the relating inelastic strains
occur with different amount because of the different expansion coefficients. Thus, the
environmental thermo-hygrometric variations produce stress and strain effects in
TCC’s. The moisture content variations also affect the timber constitutive law and
therefore cannot be neglected, along with the thermal variations, for a correct modeling
of TCC’s.
Thermal variations
The thermal properties of timber and concrete are different. Unlike concrete, timber is
in fact a good insulating material. It is interesting to evaluate the rate of variation of
temperature over the cross-section of a TCC subjected to an environmental thermal
variation T  10 °C applied with a different rate. The analyzed timber and concrete
cross-sections are those of the TCC tested in Florence (Capretti and Ceccotti 1996) and
displayed in Fig. 2. Only the portion of the concrete slab above the corrugated steel
sheet (dimensions 5x150 cm) has been considered in this analysis. The trends of
temperature in three points A, B and C chosen on the surface, at one fourth and at half
thickness of concrete flange and timber beam are displayed in Fig. 3 and 4 for the
thermal variation T applied in 1.2 hours and 0.5 day, respectively. The average value
obtained by averaging the actual temperature distribution over the cross-section at every
time t is also plotted. The results have been obtained by using a numerical program for
solving the two-dimensional heat conduction problem over the timber and concrete
cross-section with the boundary conditions of convection and irradiation.
12
Timber and concrete demonstrate a different behavior especially under quick
thermal variations. The distribution of temperature is, in fact, almost constant in the
concrete slab (Fig. 3a). The variation in time of this quantity is only slightly lower
compared to the environmental one (about 70 %) and delayed in time (about 0.35
hours). Conversely, a large difference among the temperature variations can be noted in
different points of the timber beam (Fig. 3b), where the variation in time of the average
value is low compared to the environmental one (about 35 %) and delayed in time
(about 0.5 hours). Under daily thermal variations, the aforementioned trend becomes
more marked for the concrete slab (Fig. 4a), where the temperature is constant over the
cross-section and the trend in time is nearly coincident with the environmental one. For
the timber beam, the differences among the temperatures recorded at point A, B, C
significantly reduce with respect to the hourly thermal variation (Fig. 4b). The variation
in time of the average temperature is quite large compared to the environmental one
(about 60 %) and takes place with a small delay.
On the basis of these results, a temperature variation in the concrete slab Tc
and timber beam Tw constant over the cross-section with trend in time similar to the
environmental one T may be assumed for long-term analyses of TCC’s:
Tc  y, z, t   T t 
(18)
Tw  y, z, t   kT t 
(19)
where k is a reduction factor chosen as follows:
-
For daily variations, k should be evaluated for the given timber section by solving
the heat conduction problem. For medium width timber beams such as the beam
13
tested in Florence, k may be assumed as 0.6. Higher/lower values should be used for
narrower/wider beams, respectively, with the upper/lower limits of 1/0.
-
For seasonal variations of environmental temperature (e.g. 6 months), the coefficient
k  1 can be assumed independently of the timber beam dimensions. In such a case,
in fact, the variations in time of the temperatures in timber and environment are
nearly coincident for any cross-section of technical importance.
Equations (18) and (19) represent a good approximation that allows a remarkable
simplification of the structural problem. It can be concluded that, despite the insulating
properties, timber is sensitive to thermal variations, provided that such variations are
applied slowly in time, at least on a daily scale.
Relative humidity variations
The relative humidity variations of environment affect in a different way timber and
concrete. Several properties of the former material (Young modulus, shrinkage/swelling
and mechano-sorptive creep) depend on the moisture content. Conversely, it was
observed (Capretti 1992) that for concrete even under extreme conditions, like long
cycles of absorption in water and drying in air, the relating inelastic strains are small
and take place more slowly than in timber. The influence of relative humidity changes
in time may, therefore, be neglected for concrete.
It is interesting to compare the effects in terms of moisture content variations
over a timber cross-section produced by a relative humidity variation applied on daily or
seasonal scale. A numerical program based on Eqs. (7) to (10) has been used, which
allows the solution of a two-dimensional diffusion problem over a rectangular crosssection. The comparison is performed for the glued laminated timber beam tested in
14
Florence, with reference to a single intermediate lamination (dimensions 12.5x3.3 cm).
The assumption of impermeable horizontal borders has been made, based on the remark
that the presence of the adhesive layers reduces the moisture transfer through each
couple of laminations. This approximation may be used for long-shaped cross-sections
such as that tested in Florence, where the diffusion process is mainly one-dimensional
The diffusion coefficient D of timber and the surface emissivity S of borders exposed
to the air have been assumed according to the values proposed by Toratti (1992):
D  0.10368  e ( 2.28u ) cm2/day
(20)
S  1.1232 cm/day
(21)
The trends in time of the moisture content u in three points A, B and C chosen on the
surface, at one fourth and at half width of the timber lamination are displayed in Fig. 5a
and Fig. 5b for a variation of environmental relative humidity RH  50 % applied in
half a day and in half a year, respectively. The average value computed by averaging the
actual moisture content distribution over the cross-section of the lamination at each time
t is also reported.
Daily variations of environmental relative humidity cause an uneven distribution
of moisture content over the lamination, with large fluctuations of this quantity only in
the external fibers and very small variations of average moisture content (Fig. 5a).
Seasonal variations of environmental relative humidity involve a variation of moisture
content also in the internal fibers (Fig. 5b). In both cases the fluctuation of the average
moisture content is no longer negligible and a strong difference can be recognized
among the trends in time observed in different points over the lamination. Thus, on the
15
basis of these remarks, the real local distribution of the moisture content u  u ( y, z , t )
should be considered when studying the TCC. An approximation with the trend in time
of the average value regarded as constant over the cross-section is a priori not possible.
Validation of the numerical approach
The proposed numerical procedure is validated by comparing numerical and
experimental results for the tests performed in Florence, Italy, at the Department of
Civil Engineering and in Dübendorf, Switzerland, at the EMPA laboratory. Both tests
were performed on simply supported TCC’s subjected to long-term loading.
Florence test
The cross-section of the beam tested in Florence is displayed in Fig. 2 (Capretti and
Ceccotti 1996). The concrete slab was cast on June 9, 1990 and the shores were
removed seven day after, when the dead load of 3300 N/m is considered applied. The
long-term test started on March 15, 1991, when the uniformly distributed load of 4000
N/m was applied, and continued for 5 years. The load level, quite low, corresponds to
the quasi-permanent part of the service load combination. During the test, conducted in
unsheltered well-ventilated outdoor conditions, different quantities were monitored,
such as mid-span vertical displacement, slip over the supports, timber moisture content,
temperature and relative humidity of environment. The trend in time of the maximum
and minimum daily values monitored during the 5th year of testing is displayed in Fig.
6a and 6b for the environmental temperature and relative humidity, respectively. During
the day, the temperature reaches the maximum value in the afternoon and minimum at
the sunrise, whereas the opposite happens for the relative humidity. The real
16
temperature and relative humidity histories have, therefore, been approximated by a
piecewise-linear curve connecting the maximum at 4PM and minimum at 4AM of each
day and vice versa, respectively.
Fig. 6c represents a comparison between the moisture content of timber
measured at the depth of 40 mm in a central lamination and the numerical values
obtained by solving the diffusion problem (Eqs. (7) to (10)). The numerical values have
been obtained by assuming as input the piecewise-linear relative humidity history of
environment and the values of physical properties proposed by Toratti (Eqs. (20) and
(21)). Since the timber was dried during the manufacturing process of the gluelaminated beams and the beams were exposed to the atmosphere before being connected
with the concrete slab, the hypothesis of seasoned timber has been made. Such a
hypothesis implies that timber has reached a hygrometric equilibrium with the
environment. The only drying/moistening taking place is, hence, that produced by
variations of environmental relative humidity. Numerical and experimental curves
displayed in Fig. 6c demonstrate some differences, mostly in terms of total value (about
0.03). The trends in time are similar, however the numerical solution underestimates the
experimental variations in time. Such differences may be accounted for some
uncertainties in measurement of timber moisture content and in evaluation of the timber
properties (Eqs. (10), (20) and (21)).
The long-term test has been simulated by following the whole history of the
TCC, from the concrete casting until the end of the test. Since the temperature and
relative humidity histories of environment are available only for the 5th year, the same
histories have been used also for the other years of testing. On the basis of the official
climate values recorded in the city of Florence during those years, this may be
17
considered as a reasonable approximation of the real histories. The portion of slab
inside the corrugated steel sheet (Fig. 2) has not been considered since, due to the
corrugations, there is no material continuity among them and, in addition, the
contribution on the 2nd moment of inertia is negligible. The concrete has been
considered as a viscoelastic material, according to Eqs. (4) to (6), with creep
compliance, shrinkage and Young modulus evaluated according to the CEB-FIP Model
Code 90 (C.E.B. 1993), by assuming RH  75 %, h  93.75 mm and f cm  30.43
MPa. The inelastic strains due to thermal variations have been evaluated by assuming
 c T  10  10 6 °C-1 and by considering a constant distribution of temperature in the slab
with the same trend in time as the environmental one (Eq. (18)).
The timber has been considered as a hydro-viscoelastic material, according to
Eqs. (11) to (13), with Young modulus of dried material E w0  11861 MPa. This value
corresponds to a Young modulus E w  10000 MPa, suggested by Capretti and Ceccotti
(1996), for the initial moisture content recorded at the time of application of the live
load (see the formula at denominator of the first fraction in Eq. (11)). The moisture
content distribution in time has been evaluated by solving the diffusion problem (Eqs.
(7) to (10)) with the values of diffusion coefficient and emissivity given by Eqs. (20)
and (21), and considering the timber as seasoned (see above). The material parameters
employed in Eqs. (11) and (12) have been assumed according to the values proposed by
Toratti (1992): t d  29500 days, m  0.21, u ref  0.20 , k u  1.06 , cw  2.5 and
J w  0.7 J w 0 u ref .
The
parameter
bw ,
which
takes
into
account
the
dependence
of
shrinkage/swelling on the total strain (fifth integral in Eq. (12)), was assumed equal to
18
1.3 by Toratti (1992). Such value was obtained by fitting some experimental tests
performed on timber specimens subjected to bending moment constant in time under
cycles of humidity. In TCC’s the timber is subjected to bending moment coupled with
axial force. The assumption of a non-zero value for bw would lead to non-asymptotic
trends in time of both deflections and slips for TCC’s subjected to live loads and
moisture content variations (Fragiacomo 2000). Since all long-term tests performed so
far (Capretti and Ceccotti 1996, Kenel and Meierhofer 1998) suggests the existence of
an asymptote, it is recommended that the value bw  0 be used for long-term analyses
of TCC’s. The assumption for bw of the value 1.3 would lead to small differences
during the first 5 years (1 % on maximum deflection and 10 % on maximum slip) for
the numerical simulation of the test performed in Florence, however such differences
markedly increase in the long-term ( t   ).
The moisture expansion coefficient parallel to grain  w u was not measured
during the test. In handbooks (Götz et al. 1983, Hoffmeyer 1995, Giordano 1999) the
upper limit of 10  10 3 is reported for such a quantity, independently of the type of
timber. Other authors, such as Mårtensson (1992) and Toratti (1992), report
experimental values on small spruce specimens ranging from 4.0  10 3 to 6.25  10 3 . In
this paper the value 3  10 3 has been assumed, which is closer to the value of 2.14  10 3
measured by Capretti (1992) on specimens of TCC’s made of spruce timber. The
inelastic strains due to thermal variations have been evaluated by considering a constant
distribution of temperature in the timber beam, with a trend in time similar to the
environmental one but with reduced daily variations given by Eq. (19) with k  0.6 .
The thermal expansion coefficient  w T , not measured during the test, ranges from 3 to
19
7  10 6 °C-1 for spruce timber (Götz et al. 1983, Giordano 1999), therefore the medium
value 5  10 6 °C-1 has been assumed.
The connection system was made of glued corrugated steel bars with diameter
18 mm and spacing i f from 150 mm over the supports to 300 mm up to one third of the
length and 450 mm in the middle of the span, the overall beam length being 5700 mm.
The stiffness k f , evaluated as secant value under 40 % of the shear strength in push-out
tests, was about 25000 N/mm. The connection system has been modeled as a hydroviscoelastic material, according to Eq. (14), by assuming the same material parameters
t d , m , c f , J f and b f as for timber. According to the results of some long-term pushout tests (Bonamini et al. 1990) and provisions of the prEN 1995-1-1 regulation (C.E.N.
2003) and Eurocode 5-Part 2 (C.E.N. 1996), the creep coefficient has been assumed
twice as large as the timber one, i.e. ck  2 .
The comparison between experimental and numerical results is displayed in Fig.
7a and 7b in terms of mid-span vertical displacement and slip over the support vs. time,
respectively. The numerical elastic values due to the live load are vel ,num  0.88 mm and
s f ,el ,num  0.091 mm. The experimental ones were vel ,exp  0.87 mm and s f ,el ,exp  0.045
mm. The approximation is very good in terms of deflection, less in terms of slip, which
is however a quantity rather difficult to predict. Some reasons for that are: the small
experimental value, which may imply possible measurement errors, and the influence of
both friction and adherence phenomena, which have not been modeled and may be
important especially in the case of low load level. The elastic stresses in the upper
concrete fiber and lower timber fiber due to the live load are  c ,sup  0.61 MPa and
 w,inf  0.91 MPa, which are lower than the limit values under which the viscoelasticity
20
theory and the Toratti hydro-viscoelastic model can be employed, respectively. No
cracking in concrete was observed during the whole test period, according to the
numerical simulation.
The experimental curves, obtained by averaging the mid-span deflections of the
two timber beams and the slips over the supports, do not contain the daily fluctuations.
It was taken, in fact, only one measure per day at about 4PM, the time of minimum
daily deflection and maximum slip. Conversely, the numerical curves describe a sort of
“strip” (Fig. 7a and 7b), the width of which represents the daily fluctuations. In order to
correctly evaluate the effectiveness of the numerical approach, the experimental curves
have, therefore, to be compared with the lower limit of the numerical strip for the
deflection and with the upper limit for the slip. An overall good fit can be noted,
especially during the 5th year of testing, the only one for which the real environmental
relative humidity and temperature histories are available. The numerical solution
slightly overestimates the increasing trend in time of the deflection and the relating
seasonal fluctuations. Such difference may be due to the choice of intermediate values
for the dilatation coefficients  w u and  w T , not measured during the test. The change
of  w T from 5  10 6 °C-1 to 7  10 6 °C-1 and  w u from 3  10 3 to 2.14  10 3 , in the
range of possible values, would lead to an appreciable reduction in the seasonal
fluctuation (about 30 %). Conversely, the choice of  w T  3  10 3
°C-1 and
 w u  6.25  10 3 would maximize the seasonal fluctuation (about 55 % more). The slips
are instead slightly underestimated by the numerical solution, but a good coincidence
can be noted in terms of seasonal fluctuations. Variations similar to those computed for
the deflection would take place by assuming the upper and lower limit values for the
hygrometric and thermal expansion coefficients. The comparison between experimental
21
and numerical daily fluctuations (Fragiacomo 2000) has also provided good results,
especially in terms of deflection. The proposed numerical approach can, therefore,
adequately fit with the experimental deflection and slip measured in the long-term test
performed in Florence.
EMPA test
The TCC “B5” tested under long-term loading in sheltered outdoor conditions at the
EMPA laboratory (Kenel and Meierhofer 1998) is displayed in Fig. 8. Such specimen
was characterized by a connection system made of couples of SFS-screws 45° inclined
with respect to the beam axis, with an average stiffness k f  17000 N/mm for each
couple of screws. The spacing i f between each couple of screws was 100 mm from the
support to the point of abscissa x  72.5 cm, 133 mm from that point to one third of the
length and 300 mm in the middle third of the span, the span length being 3850 mm.
The shores were removed 21 days after the concrete casting and the dead load
was g  1700 N/m. Two concentrated live loads Q  6000 N were applied at one and
two thirds of the span 28 days after the concrete casting, when the monitoring of midspan deflection began. The mechanical characteristics of the component materials were:
concrete
f cm  34.8 MPa, timber E w0  14846 MPa, which corresponds to the
measured Young modulus E w  12800 MPa for a moisture content of about u  0.13 .
The creep properties of timber and concrete, not measured during the test, have been
modeled using the Toratti and the CEB-FIP Model Code 90 approaches respectively, by
assuming h  132 mm and RH  75 % for the evaluation of creep and shrinkage of
concrete. The creep compliance of connection system was determined through push-out
22
tests under long-term loading, from which the authors (Kenel and Meierhofer 1998)
proposed the function:
J f (t , ) 

1
n
1  d t  
Kf

(22)
with d  0.117 and n  0.279 . This function has been approximated by means of the
Kelvin generalized rheological model (Eq. (14)), by assuming c f  1 and by calibrating
the coefficients  f n and J f n in order to obtain the best fit. The mechano-sorptive creep
has been modeled for both timber and connection by using the Toratti model with the
same parameters adopted in the previous Section. Neither the moisture expansion
coefficient  w u nor the thermal expansion coefficients  c T ,  w T were measured, thus
the medium values 3  10 3 , 10  10 6 °C-1 and 5  10 6 °C-1 have been assumed,
respectively.
The environmental relative humidity and temperature histories were not
monitored during the test. Thus the approximate sinusoidal temperature history
displayed in Fig. 9a has been considered, which has been obtained on the basis of the
official climate values. Since no information on environmental relative humidity is
available, the actual moisture content distribution on the cross-section cannot be
computed. The long-term numerical solution has then been obtained by using the timber
moisture content history measured during the test (Fig. 9b), regarding such a quantity as
constant over the cross-section. Despite such rather crude approximations, the
prediction of the experimental mid-span deflection is quite good (Fig. 10), with the
numerical model slightly underestimating the experimental response. The elastic values
23
are very close ( vel ,num  2.78 mm compared to vel ,exp  2.66 mm) and the seasonal
fluctuations are well approximated. The elastic stresses in the upper concrete fiber and
lower timber fiber due to the live loads are  c ,sup  2.87 MPa and  w,inf  3.43 MPa
respectively, which enable the assumptions of viscoelastic concrete and hydroviscoelastic timber. Cracking in the concrete slab has been observed neither in the
numerical simulation nor in the experimental test. The deflection monitored in the
EMPA test can, therefore, be fitted with a good accuracy by the proposed approach.
Conclusions
The first part of two companion papers deals with the numerical modeling of TimberConcrete Composite beams (TCC’s) under long-term loading. The procedure is based
on a uniaxial finite element model that considers the flexibility of connection system
and the time-dependent behavior of timber, concrete and connection system. Creep,
mechano-sorptive creep, shrinkage/swelling and thermal variations are taken into
account by using accurate linear models. In order to obtain an effective computational
process, the creep compliances are transformed in sums of exponential functions. The
solution is obtained by applying a step-by-step integration procedure over time.
The environmental thermo-hygrometric variations affect the behavior of TCC’s
since they causes inelastic strains in timber and concrete, influence the elastic modulus
of timber and the mechano-sorptive creep of both timber and connection system. The
temperature can be considered as constant over the cross-section of timber and concrete.
The trend in time of such quantity may be assumed the same as the environmental one
for concrete. For timber, a trend in time similar to the environmental one with same or
reduced amplitude (40 % less for medium width timber sections) may be assumed,
24
respectively, for seasonal or daily variations. The moisture content of timber, instead, is
characterized by an uneven distribution over the cross-section. Because of its important
influence on the timber behavior, i.e. on mechano-sorptive creep, Young modulus and
shrinkage/swelling, it has to be evaluated by solving the diffusion problem once the
relative humidity history of environment is known. The comparison between numerical
and experimental moisture content recorded in the test performed in Florence has
pointed out similar trends but differences on variations in time and, mostly, on total
values.
The numerical procedure for long-term analysis of TCC’s has been validated on
two experimental tests. The comparisons carried out have pointed out a quite good
approximation for the deflections, even if i) the diffusion analysis delivers some
differences with the experimental moisture content or if ii) a rather crude approximation
of moisture content constant over the whole cross-section is assumed. The elastic values
are very well estimated and the differences on the delayed values and on the seasonal
fluctuations are small. Such differences may be justified because of the approximations
on the choice of expansion coefficients, environmental temperature and relative
humidity histories. The slightly larger difference between experimental and numerical
solutions in terms of slip over the support may be justified by the small values recorded
in the test, which are due to the low load level (the quasi-permanent part of the service
load) and by some local phenomena, such as friction and adherence between concrete
and timber, which cannot be adequately modeled. The proposed numerical procedure
can, therefore, be employed to predict the long-term behavior of TCC’s under variable
environmental conditions.
25
The model is used in the second part to perform a parametric analysis showing
the contribution of every single rheological phenomenon and thermo-hygrometric
variations on beam deflection and connection slip. Based on results carried out, a
simplified approach for long-term analyses of TCC’s is then proposed. Closed form
solutions are employed to evaluate effects of dead and live loads, concrete shrinkage
and inelastic strains due temperature and moisture content variations. The accuracy of
the proposed method is demonstrated through some comparisons.
References
Amadio, C., and Fragiacomo, M. (1993). “A finite element model for the study of creep
and shrinkage effects in composite beams with deformable shear connections.”
Costruzioni Metalliche, 4, 213-228.
Amadio, C., Fragiacomo, M., Ceccotti, A., and Di Marco, R. (2001). “Long-term
behaviour of a timber-concrete connection system.” Proc., RILEM Conference
“Joints in timber structures”, Sept. 12-14, Stuttgart, Germany, 263-272.
Ballerini, M., Crocetti, R., and Piazza, M. (2002). “An experimental investigation on
notched connections for timber-concrete composite structures.” Proc., The 7th World
Conference on Timber Engineering, WCTE 2002, Aug. 12-15, Shah Alam, Malaysia,
2, 171-178.
Balogh, J., Gutkowski, R.M., Wieligmann, M., and Haller, P. (2002). “Mechanics
behavior of dowel connections for partially composite wood-concrete beams.”
Proc., The 7th World Conference on Timber Engineering, WCTE 2002, Aug. 12-15,
Shah Alam, Malaysia, 3, 290-295.
26
Bonamini, G., Ceccotti, A., and Uzielli, L. (1990). “Short- and long-term experimental
tests on antique larch and oak wood-concrete composite elements.” Proc., C.T.E.
Conference, Bologna, Italy, 241-251 (in Italian).
Capretti, S. (1992). “Time dependent analysis of timber and concrete composite (TCC)
structures.” Proc., RILEM International Symposium on “Behaviour of timber and
concrete composite load-bearing structures”, June 27, Ravenna, Italy.
Capretti, S., and Ceccotti, A. (1996). “Service behaviour of timber-concrete composite
beams: a 5-year monitoring and testing experience.” Proc., International Wood
Engineering Conference, New Orleans, USA, 3, 443-449.
Comité Euro-International du Béton (1993). “CEB-FIP Model Code 90.” CEB Bull. No.
213/214, Lausanne, Switzerland.
Comité Européen de Normalisation (1995). “Eurocode 5 – Design of Timber Structures
– Part 1-1: General Rules and Rules for Buildings.” ENV 1995-1-1, Bruxelles,
Belgium.
Comité Européen de Normalisation (1996). “Eurocode 5 – Design of Timber Structures
– Part 2: Bridges.” ENV 1995-2, Bruxelles, Belgium.
Comité Européen de Normalisation (2003). “Eurocode 5 – Design of Timber Structures
– Part 1-1: General Rules and Rules for Buildings.” prEN 1995-1-1, Bruxelles,
Belgium.
Fragiacomo, M. (2000). “Long-term behavior of timber-concrete composite beams.”
Ph.D. Thesis, University of Trieste, Italy (in Italian).
Giordano, G. (1999). Technique of timber constructions. Hoepli Editore S.p.A., Milan,
Italy (in Italian).
Götz, K.-H., Hoor, D., Möhler, K., and Natterer, J. (1983). Costruire en bois: Choisir,
27
Concevor, Réaliser. Editions du Moniteur, Presses Polytechniques Romandes,
Lausanne, Switzerland (in French).
Hanhijärvi, A., and Hunt, D. (1998). “Experimental indication of interaction between
viscoelastic and mechano-sorptive creep.” Wood Science and Technology, 32, 57-70.
Hoffmeyer, P. (1995). “Wood as a building material.” Timber Engineering, Step 1, First
Edition, Centrum Hout, The Netherlands, A4/1-A4/21.
Kenel, A., and Meierhofer, U. (1998). “Long-term performance of timber-concrete
composite structural elements.” Report No. 115/39, EMPA, Dübendorf, Switzerland
(in German).
Kuhlmann, U., and Schänzlin, J. (2001). “Composite of vertically laminated timber
decks and concrete.” Proc., IABSE Conference “Innovative wooden structures and
bridges”, Lahti, Finland, 507-512.
Lacidogna, G. (1994). “Mathematical modeling of the viscoelastic behavior of
concrete.” Ph.D. Thesis, Polytechnic of Turin, Italy (in Italian).
Mårtensson, A. (1992). “Mechanical behaviour of wood exposed to humidity
variations.” Report TVBK-1006, Lund Institute of Technology, Lund, Sweden.
Newmark, N.M., Siess, C.P., and Viest, I.M. (1951). “Tests and analysis of composite
beams with incomplete interaction.” Proc., Society for Experimental Stress Analysis,
9(1), 75-92.
Ranta Maunus, A. (1975). “The viscoelasticity of wood at varying moisture content.”
Wood Science and Technology, 9, 189-205.
Said, E.B., Jullien, J.-F., and Siemers, M. (2002). “Non-linear analyses of local
composite timber-concrete behaviour.” Proc., The 7th World Conference on Timber
Engineering, WCTE 2002, Aug. 12-15, Shah Alam, Malaysia, 1, 183-191.
28
Toratti, T. (1992). “Creep of timber beams in a variable environment.” Report No. 31,
Helsinki University of Technology, Helsinki, Finland.
29
Figure Captions
Fig. 1. Finite element
Fig. 2. Cross-section of the TCC tested in Florence (measures in cm)
Fig. 3. Trend in time of temperature in three points of the concrete slab (a) and timber
beam (b) for an environmental thermal variation of 10 °C applied in 1.2 hours
Fig. 4. Trend in time of temperature in three points of the concrete slab (a) and timber
beam (b) for an environmental thermal variation of 10 °C applied in half a day
Fig. 5. Trend in time of moisture content in three points of the timber lamination under
a variation of environmental relative humidity of 50 % applied in half a day (a) and in
half a year (b)
Fig. 6. Trend in time of maximum and minimum environmental daily temperature (a),
maximum and minimum environmental relative humidity (b), experimental and
numerical timber moisture content (c) during the 5th year of testing
Fig. 7. Comparison between numerical and experimental trend in time of the mid-span
vertical displacement (a) and slip over the support (b) for the Florence test
Fig. 8. Cross-section of the TCC tested at the EMPA laboratory, Dübendorf (measures in
cm)
Fig. 9. Adopted environmental temperature history (a) and measured timber moisture
content history (b) for the EMPA test
Fig. 10. Comparison between numerical and experimental trend in time of the mid-span
vertical displacement for the EMPA test
30
Concrete
Concrete
slab slab
Ar1
Ar1
Gc Ar2Gc Ar2
H
H
c
c
Nucc
c
Mc
Gw
z
z
Nc
Gw
Mw
Rigid links
Rigid links
Nc+dNcNc+dNc
uc
c
vc
Mc vc
c
c
Kf(x) Kf(x)
Nw
N
uww
uw
Mw vw
y
y
Timber Timber
beam beam
Mc+dMM
c c+dMc
x
x
Nw+dNwNw+dNw
vw
dx
Mw+dMM
ww+dMw
dx
Fig. 1. Finite element
steel mesh  6 size 30x30 cm
concrete slab
2.5
5.0
10.0
corrugated steel sheet
fastener
50.0
glulam beam
27.5
12.5
70.0
12.5
27.5
Fig. 2. Cross-section of the TCC tested in Florence
(measures in cm)
31
22
22
T [°C]
T [°C]
C A B
20
20
Environment
Environment
18
18
16
16
A
B
Average
14
t [hours]
0.6
1.2
1.8
C
B
t [hours]
10
2.4
B
Average
12
10
0
C
A
14
C
12
A
0
0.6
1.2
1.8
2.4
(a)
(b)
Fig. 3. Trend in time of temperature in three points of the concrete slab (a) and timber beam (b) for an
environmental thermal variation of 10 °C applied in 1.2 hours
22
T [°C]
20
22
C A B
A, B, C and average
T [°C]
20
18
18
16
16
14
14
12
12
Environment
A
Environment
A
C
B
C
Average
t [days]
10
0
0.5
1
1.5
2
2.5
B
t [days]
10
3
0
0.5
1
1.5
2
2.5
3
(a)
(b)
Fig. 4. Trend in time of temperature in three points of the concrete slab (a) and timber beam (b) for an
environmental thermal variation of 10 °C applied in half a day
32
0.24
B
B
u [-]
RH-environment
0.20
A
0.24
RH [%]
u [-]
100
C
0.20
RH [%]
B
B
RH-environment
C
80
0.16
0.16
u at A
60
0.08
0
0.5
1
1.5
2
u-average
40
t [days]
2.5
3
60
u at B
0.12
u at B and C
100
80
u at C
u at A
u-average
0.12
A
40
t [days]
0.08
0
91.25
182.5
273.75
365
(a)
(b)
Fig. 5. Trend in time of moisture content in three points of the timber lamination under a variation of
environmental relative humidity of 50 % applied in half a day (a) and in half a year (b)
33
40
30 T [°C]
20
10
0
3/15/1995
-10
0
60
t [days]
120
180
240
300
360
(a)
100
80
60
40
20 RH [%]
0
0
60
t [days]
120
180
240
300
360
(b)
0.22
u [-] Numerical-40 mm depth
0.20
0.18
0.16
0.14
Experimental-40 mm depth
0.12
0
60
120
180
Numerical-average
t [days]
240
300
360
(c)
Fig. 6. Trend in time of maximum and minimum
environmental daily temperature (a), maximum and
minimum environmental relative humidity (b),
experimental and numerical timber moisture content
(c) during the 5th year of testing
34
5
0.5
v [mm]
Experimental
4
sf [mm]
Experimental
0.4
3
Numerical
0.3
2
0.2
Numerical
1
0.1
t [years]
0
t [years]
0
0
1
2
3
4
5
0
1
2
3
4
5
(a)
(b)
Fig. 7. Comparison between numerical and experimental trend in time of the mid-span vertical
displacement (a) and slip over the support (b) for the Florence test
75
Steel mesh 5 size 10x10 cm
Concrete slab
4
4
2
Timber decking
SFS screws connection
18
Timber beam
12
Fig. 8. Cross-section of the TCC tested at the
EMPA laboratory, Dübendorf (measures in cm)
35
40
30
20
10
0
-10
t [years]
0
1
2
3
4
5
(a)
0.18
0.17
0.16
0.15
0.14
0.13
u [-]
t [years]
0
1
2
3
4
5
(b)
Fig. 9. Adopted environmental temperature
history (a) and measured timber moisture content
history (b) for the EMPA test
12
v [mm]
10
8
6
4
Numerical
Experimental
2
t [years]
0
0
1
2
3
4
5
Fig. 10. Comparison between numerical and
experimental trend in time of the mid-span vertical
displacement for the EMPA test
36