Long-Term Behavior of Timber-Concrete Composite Beams.
II: Numerical Analysis and Simplified Evaluation
Massimo Fragiacomo1
Abstract: The second part of two companion papers investigates the contribution of
different rheological phenomena and thermo-hygrometric variations on long-term
behavior of Timber-Concrete Composite beams (TCC’s) in outdoor conditions. The
numerical algorithm presented and validated against two experimental tests in the first
part is employed with this aim. Such a model fully considers all rheological phenomena
and, therefore, leads to rigorous solutions. Effects on the beam response include the
creep and mechano-sorptive creep of both timber and connection, along with concrete
creep and shrinkage, and may markedly increase the elastic deflection due to live load.
The inelastic strains due to yearly and daily variations of environmental conditions
(temperature and relative humidity) produce an important fluctuation of the deflection.
A simplified method, which is suitable for practical design of TCC’s under long-term
loading, is at last proposed. The effects of load, concrete shrinkage and inelastic strains
due to environmental variations are evaluated one by one using approximate formulae
and are then superimposed. Creep and mechano-sorptive creep are taken into account by
adopting modified elastic moduli. The reliability of the proposed method is checked by
way of some comparisons with numerical results. The applicability for the case of
TCC’s in heated indoor conditions is also discussed.
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.
1
CE Database keywords: Composite beams; Concrete; Creep; Rheological
properties; Time dependence; Shrinkage; Timber constructions; Wood.
Introduction
The Timber-Concrete Composite beam (TCC) is a structural technique employed for
both new constructions and the restoration of existing buildings. The presence of a
concrete flange linked to the timber beam through a connection system allows a good
performance in terms of both strength and stiffness. A further benefit lies in the fact that
the composite floor behaves as a rigid diaphragm, which is particularly important for a
good structural performance in seismic regions.
Both ultimate and serviceability limit states have to be satisfied, according to
design codes (C.E.N. 1995, 1996). Because of the rheological phenomena, such as creep
and mechano-sorptive creep, and because of shrinkage/swelling of the component
materials, the stress and strain distribution change in time, and the verifications have to
be carried out at least in two stages: (i) the time of loading (short-term verification); (ii)
the end of the service life, usually 50 years for domestic or office buildings (long-term
verification). The rheological phenomena have to be taken into account for the longterm controls. Few investigations were carried out in order to estimate such
contributions. Mola et al. (1989) solved the problem by assuming a linear-viscoelastic
behavior, whereas Ceccotti and Covan (1990) employed a numerical approach based on
conventionally time-reduced elastic moduli for concrete, timber and connection.
Mungwa and Kenmou (1993) used the finite difference method in order to solve the
viscoelastic problem for TCC’s with rigid connection. Other numerical approaches were
proposed by Capretti (1992), Kuhlmann and Schänzlin (2001), and Said et al. (2002).
2
In the first part of the two companion papers (Fragiacomo and Ceccotti 2005) a
uniaxial finite element model with flexible connection has been proposed for long-term
analyses of TCC’s. All phenomena affecting the long-term behavior of timber, concrete
and connection system, such as creep, mechano-sorptive creep, shrinkage/swelling and
temperature variations, have been fully considered. The moisture content distribution
over the timber cross-section has been evaluated by solving the diffusion problem. The
important influence of environmental thermo-hygrometric variations for beams in
outdoor conditions has been recognized. The proposed numerical model which leads to
rigorous solutions has been validated against two long-term experimental tests.
The second part investigates the contribution of various rheological phenomena
and thermo-hygrometric variations on both deflection and slip of TCC’s exposed to
outdoor conditions. The analyses are performed using the numerical model developed in
the first part for two TCC’s subjected to loading and environmental conditions
monitored during the tests and extended to the entire service life. Most of all, shrinkage
and creep of concrete, creep and mechano-sorptive creep of timber and connection
increase the vertical deflection and, therefore, should not be neglected. Both yearly and
daily variations of environmental temperature and relative humidity produce a change in
the strain and stress state of the beam. The importance of these contributions and the
identification of the most influencing parameters are also discussed.
The need for simplified approaches to design TCC’s is particularly important in
order to promote the use of this technique. The classical approach, proposed by Ceccotti
(1995) according to Eurocodes 5 – Parts 1-1 and 2 (C.E.N. 1995, 1996), is based on the
use of the formulae for TCC’s with flexible connections (Möhler 1956). In this
approach, the effect of creep in long-term verifications is taken into account by applying
3
the Effective Modulus Method (Chiorino et al. 1984). Conversely, concrete shrinkage
and change of environmental conditions are not considered. Since experimental tests
and numerical analyses pointed out that these effects may be important, a simplified
evaluation is proposed. The total solution is obtained by superimposing the hydroviscoelastic solutions due to loading and concrete shrinkage, and the elastic solutions
due to yearly and daily thermo-hygrometric environmental variations. Numerical and
approximate solutions are then compared. The simplicity and accuracy of prediction
obtained make the proposed method suitable for practical design of TCC’s. The
simplifications for the case of TCC’s in heated indoor conditions are also discussed.
Numerical analysis
The component materials of TCC’s, i.e. timber, concrete and connection, demonstrate
important rheological phenomena, which affect the structural behavior under long-term
loading. If a hydro-viscoelastic model, such as the Toratti one (1992), is employed, the
following constitutive law can be written for each component material:
t    J 0 (u ()) d   J c (t , ) d    dJ 0 u    J ms (U ()) d 
t
t
t0
t
t0
t
t0
t0
  bdu     u du     T dT    d s 
t
t
t
t
t0
t0
t0
t0
(1)
where J 0 u()   1 / E (u()) , J c t ,   (t , ) / E (u ref ) , U ()   du(1 ) , and  , J ms ,
t

E , u , u ref , b ,  u ,  T ,  s ,  ,  , t 0 , t ,  are, respectively, creep coefficient,
mechano-sorptive creep compliance, Young modulus, moisture content, a reference
value of the moisture content, a material parameter assumed as zero (Fragiacomo and
Ceccotti 2005), moisture expansion coefficient, thermal expansion coefficient, concrete
4
shrinkage strain, total strain, stress, initial time, final time and current time of analysis.
Each integral in Eq. (1) represents a different contribution on the total strain:
   e   c   e,u   ms   u , s   u   T   s
(2)
where  e is the elastic strain and  c ,  ms ,  e,u ,  u, s ,  u ,  T ,  s are, respectively, the
strain components due to creep, mechano-sorptive creep, dependence of the Young
modulus on the moisture content, shrinkage/swelling dependent (subscript “u,s”) and
independent (sub. “u”) of the total strain  caused by moisture content variations,
shrinkage/swelling caused by temperature variations, and shrinkage of concrete. Eq. (2)
can be rewritten for each material (concrete, subscript “c”, timber, sub. “w”, and
connection, where the slip s f replaces the strain  ) by simplifying the zero terms:
c  c e  c c  cT  c s
(3)
 w   w e   w c   w e,u   w ms   w u   w T
(4)
s f  s f e  s f c  s f ms
(5)
The strain/slip components  c c ,  w c ,  w e,u ,  w ms , s f c and s f ms depend on stress  and
hence are considered as “rheological phenomena”, i.e. as part of the constitutive law.
The strains  w u ,  w T ,  c T and  c s are independent of both stress  and total strain  ,
thus are defined as “inelastic strains”. Since their histories depend only on thermohygrometric environmental variations, they can be regarded as loading conditions. The
use of linear models with respect to stress, such as the viscoelasticity theory for concrete
5
(Eq. (3)) and the hydro-viscoelasticy (Toratti) theory for timber and connection (Eqs.
(4) to (5)), enables the use of the superposition principle. Hence the global long-term
solution of TCC’s can be obtained by superimposing the following loading conditions:
-
dead and live load;
-
concrete shrinkage (  c s );
-
shrinkage/swelling of concrete (  c T ) and timber (  w T ) due to thermal variations,
and shrinkage/swelling of timber due to moisture content variations (  w u ),
herein after referred to as “inelastic strains due to environmental variations”.
In order to study the influence of the different rheological phenomena on TCC’s,
the three loading conditions are studied one by one by means of the finite element
model described in the first part of the two companion paper (Fragiacomo and Ceccotti
2005). The numerical simulations have been carried out for two different simply
supported TCC’s, which were tested in outdoor conditions by Capretti and Ceccotti
(1996) at the University of Florence and by Kenel and Meierhofer (1998) at the EMPA
laboratory. Herein after, such beams will be referred to as “Florence beam” and “EMPA
beam”. Geometrical, mechanical and rheological properties of the beams are
summarized in Table 1. The numerical analyses are carried out for the entire service life
of the structure, estimated as 50 years. For the Florence beam, environmental
temperature and relative humidity histories monitored during the 5th year of testing (Fig.
1a and 1b) have been extended to the whole period of analysis. Both histories are
assumed to linearly change between the maximum and minimum daily values. For the
EMPA beam, environmental temperature and average moisture content of timber during
the 5 years of testing (Fragiacomo and Ceccotti 2005) have also been extended to the
whole period of analysis. No daily variation has been considered in this case.
6
Load and concrete shrinkage
The responses to the loading condition of only concrete shrinkage (lower curves) and to
the loading condition given by both live load and concrete shrinkage (upper curves) are
displayed in Fig. 2 in terms of mid-span vertical displacement and slip over the support
for the Florence beam, and in Fig. 3 in terms of mid-span vertical displacement for the
EMPA beam. The effect of only live load (  c s  0 in Eq. (3)) can also be recognized as
the difference between upper and lower curves. Deflection and slip are plotted as the
ratio between current and elastic values due to the live load. Each figure contains
several curves, which refer to different constitutive laws adopted for the component
materials:
a) hydro-viscoelastic connection (Eq. (5)), hydro-viscoelastic timber with Young
modulus depending on the moisture content (Eq. (4) with  w u  0 and  w T  0 ) and
viscoelastic concrete (Eq. (3) with  c T  0 );
b) same as a) but with viscoelastic connection (Eq. (5) with s f ms  0 );
c) same as b) but with viscoelastic timber (Eq. (4) with  w ms  0 ,  w u  0 and  w T  0 );
d) same as c) but with elastic connection (Eq. (5) with s f ms  0 and s f c  0 );
e) same as d) but with elastic timber (Eq. (4) with  w c  0 ,  w ms  0 ,  w u  0 and
 w T  0 );
f) same as e) but with elastic concrete (Eq. (3) with  c c  0 and  c T  0 );
g) elastic materials with Young modulus of timber independent of the moisture content
(Eq. (4) with  w e ,u  0 ,  w c  0 ,  w ms  0 ,  w u  0 and  w T  0 ).
The difference between two consecutive curves may be assumed, approximately, as a
measure of influence of the relating rheological phenomenon on the beam response. In
7
all analyses except the last one (curve “g”), the history of moisture content u  uP, t  ,
as evaluated by solving the diffusion problem for the environmental relative humidity
history RH  RH (t ) (Fig. 1b), has been considered for the Florence beam in each point
P of the timber cross-section. For the EMPA beam, the average moisture content
history monitored during the experimental test has been used. The comparison between
numerical (hydro-viscoelastic solution, curve “a”) and experimental values of deflection
and slip monitored during the 5th year of testing is reported in Table 2 and in Figs. 2 to 3
for both beams, and demonstrates the accuracy of the FE model. The contributions of
different loading conditions on maximum numerical values obtained during the entire
service life (50 years) are also reported in the same table, as deflection/slip to span
length ratios and as percentages of the elastic values due to live load. Table 3
summarizes the influence of different rheological phenomena on final deflection and
slip of both beams by distinguishing between live load and concrete shrinkage. The
following remarks can be made for the numerical analyses carried out:
-
The live load always increases deflection and slip, whereas the concrete shrinkage
increases deflection and decreases slip (Figs. 2, 3 and Tab. 2). The trend of the total
slip, therefore, depends on the balance between these two contributions (Fig. 2b).
-
The creep and mechano-sorptive creep phenomena raise the elastic deflection and
slip due to the live load (Figs. 2, 3 and Tab. 3). All of them, separately considered,
augment both deflection and slip (except the concrete creep for the slip), as can be
noted by comparing the different curves from “a” to “f”.
-
The creep and mechano-sorptive creep phenomena increase the elastic slip (as an
absolute value) and reduce the elastic deflection (curves “g”) due to the concrete
shrinkage (Figs. 2, 3 and Tab. 3). The rheological phenomena of timber increase the
8
deflection and decrease the slip (as an absolute value), whereas the opposite is
produced by the rheological phenomena of the connection.
-
The larger value of deflection produced by the concrete shrinkage on the EMPA
beam (Fig. 3 and Tab. 2), about 3.4 times the value of the Florence beam, is mainly
due to the lower initial time of monitoring. However, the response of the beam is
also influenced by several other parameters such as environmental conditions,
geometry and stiffness of the component materials.
-
The contributions of concrete creep, creep and mechano-sorptive creep of timber
and connection on the global deflection and slip are comparable. The increasing
trend of slip (curve “a” in Fig. 2b), also recorded in the experimental test performed
in Florence, may be justified only taking into account the contribution of mechanosorptive creep of timber and connection. If these two phenomena had been neglected
(curve “c”), the trend would have been a decreasing one during the first 5 years.
-
The dependency of the timber Young modulus on the moisture content implies only
light ripples on the curves and is negligible, as can be noted by Table 3 and by
comparing the curves “g” and “f” in Figs. 2 and 3.
-
The overall effect of rheological phenomena and concrete shrinkage is remarkable:
they lead (Fig. 2, 3 and Table 2) to a final deflection equal to 492 % and 434 % of
the elastic deflection due to live load for the Florence and EMPA beam,
respectively.
Inelastic strains due to environmental variations
The inelastic strains to be considered are (Fragiacomo and Ceccotti 2005) those due to
moisture content variations of timber (  w u in Eq. (4)), thermal variations of timber
9
(  w T in Eq. (4)) and concrete (  c T in Eq. (3)). The moisture content history u  u ( P, t )
is evaluated for the Florence beam at each point P of the timber cross-section by
solving the diffusion problem in relation to the environmental relative humidity history
RH  RH t  (Fig. 1b). The concrete and timber temperatures, Tc and Tw respectively,
are regarded as constant over each cross-section and variable in time with trend similar
to the environmental one T  T (t ) , which fluctuates with daily period between the two
limit curves displayed in Fig. 1a. The daily amplitudes assumed for Tc and Tw are
Tc  Tdaily and Tw  kTdaily , where Tdaily is the daily variation of environmental
temperature. The parameter k is a reduction factor depending on geometry and physical
properties of the timber beam, equal to 0.6 for such a beam. The daily fluctuations also
occur for the average timber moisture content, but with very small amplitude (Fig. 1c).
Furthermore, an important yearly fluctuation can be recognized for both quantities. The
temperature reaches a maximum in August and a minimum in December, with a yearly
variation T yearly  28 °C (Fig. 1a), whereas the opposite occurs for the average timber
moisture content, with a yearly variation u yearly  0.035 (Fig. 1c).
The solutions for inelastic strains due to environmental variations, obtained by
means of a rigorous numerical hydro-viscoelastic analysis (curves “a” - Eqs. (3) to (5)
with  c s  0 ) are plotted in Fig. 4 for the Florence beam and in Fig. 5 for the EMPA
beam, for the 50th year of analysis (right ordinates). The daily fluctuations, recognizable
as width of the “strips” described by the rigorous numerical curves (Fig. 4), are mainly
caused by daily temperature variations. The yearly fluctuations are caused by both
yearly temperature and moisture content variations. An increase in temperature
produces a dilatation of both the concrete slab and timber beam, but the former
10
contribution prevails because of the larger thermal expansion coefficient of concrete,
leading to negative (upward) deflection and positive slip. A decrease in timber moisture
content produces similar effects. The cooling from day to night, therefore, produces
downward deflection ( v daily  0.43
mm for medium temperature differences
Tdaily  8 °C), negative slip ( s f ,daily  0.05 mm), positive bending moments and
negative axial forces. The cooling from summer to winter, along with the moistening of
timber, leads to similar but larger effects. Altogether, daily and yearly environmental
variations produce important effects on the TCC, since the maximum fluctuations are
v  2.24 mm and s f  0.258 mm, equal to 254 % and 284 % of the elastic
deflection and slip due to the live load, respectively. Similar results have been obtained
for the EMPA beam (Fig. 5), where the fluctuation of deflection due only to yearly
variations is v  1.65 mm, which is 59 % of the elastic deflection due to the live load.
The inelastic strains due to environmental variations cause, therefore, important effects
in TCC’s working in outdoor conditions.
It is also interesting to evaluate the effects of creep and mechano-sorptive creep
on the solutions due to environmental variations. The numerical comparison between
the rigorous hydro-viscoelastic analysis (Eqs. (3) to (5) with  c s  0 ) and a linearelastic analysis (Eqs. (3) to (5) with  c c  0 ,  c s  0 ,  w e ,u  0 ,  w c  0 ,  w ms  0 ,
s f c  0 and s f ms  0 ) reveals only small differences (7.8 % of elastic deflection and 15
% of elastic slip due to the live load for the Florence beam). This result can be justified
by the cyclic trend in time of the solution, which makes the effects of creep phenomena
negligible. It may be concluded that the solution for inelastic strains due to
environmental variations can be approximately evaluated through an elastic analysis.
11
Superposition
In order to obtain the maximum effects during the service life of the beam, only a part
of the fluctuation due to environmental variations has to be superimposed to the
maximum values due to concrete shrinkage and load. The maximum effects due to
environmental variations are influenced by the time of concrete hardening t 0 with
respect to the times t w of minimum temperature Tmin and maximum moisture content
u max , and t s , time of maximum temperature Tmax and minimum moisture content u min .
The time of concrete hardening t 0 can be estimated as 1 to 3 days after casting, since in
such a time concrete changes from plastic to solid (hardened) state and develops its own
strength. In order to evaluate maximum deflection and bending moments, the following
environmental variations have to be considered:
T  Tmin  T t 0  with Tmin  T t w 
(6)
uP   u max P   uP,t 0  with u max P   uP, t w 
(7)
whereas to evaluate maximum slip and axial forces the quantities Tmin , u max and t w in
Eqs. (6) to (7) have to be replaced by Tmax , u min and t s . The maximum deflection and
bending moments, therefore, are obtained if t 0  t s , whereas the maximum slip and
axial forces occur if t 0  t w .
Since the beam tested in Florence was monitored from the time of loading t Q ,
which was in spring during afternoon, the maximum values of deflection and slip
monitored during the test are influenced by such a quantity instead of the time of
12
concrete hardening t 0 . In order to obtain the maximum deflection, which occurs in
winter during the night, the whole daily variation and a part of the yearly fluctuation
have to be taken into account. Conversely, no daily variation has to be considered for
the maximum slip, which occur in summer during the day. For the EMPA beam, only a
little part of the yearly fluctuation has to be taken into account since the times t Q and t w
are nearly coincident. No daily variation has been considered since no experimental data
is available. The maximum values obtained are reported in Figs. 4 to 5 and Tab. 2 along
with the total values calculated by superimposing the different contributions. The
rheological phenomena, concrete shrinkage and inelastic strains due to environmental
variations produce altogether a remarkable increase in deflection and slip with respect to
the elastic values due to live load, leading to final values of 451 % for the EMPA beam,
635 % and 442 % for the Florence beam, respectively. The most important contribution
is the delayed effect due to vertical load, however the effects of concrete shrinkage and
inelastic strains are important as well and cannot be neglected.
Simplified evaluation
The aim of this Section is to provide a simplified method for long-term analysis of
TCC’s employable for practical design. The rigorous solution, in fact, involves the use
of sophisticated numerical algorithms, such as those developed by Capretti (1992),
Kuhlmann and Schänzlin (2001), Said et al. (2002), Fragiacomo and Ceccotti (2005),
which are generally too complex and not suitable for practical design. The present
approach is an extension of that proposed by Ceccotti (1995) according to Eurocodes 5
– Parts 1-1 and 2 (C.E.N. 1995, 1996), in order to account also for concrete shrinkage
and inelastic strains due to environmental variations.
13
Proposed procedure
According to the previous Section, the long-term solution for TCC’s in outdoor
conditions can be obtained by superimposing the effects of dead and live loads, concrete
shrinkage and inelastic strains due to environmental variations. The yearly trends of
environmental temperature and average timber moisture content are approximated by
piecewise-linear curves (Figs. 1a and 1c for the Florence beam). Such quantities are
considered as constant over the timber and concrete cross-sections. A daily variation of
temperature is considered as well. The dependency of the timber Young modulus on the
moisture content and the daily variations of average moisture content are neglected. The
effects of inelastic strains due to environmental variations are evaluated through an
elastic analysis. The effects of creep and mechano-sorptive creep, on solutions due to
loads and concrete shrinkage, are evaluated by using the Effective Modulus Method
(Chiorino et al. 1984). The elastic solution of TCC’s with flexible connection subjected
to loads is evaluated through the simplified formulae proposed by Möhler (1956), the
so-called “gamma-method”. Such formulae are widely employed with some
modifications in timber engineering also for three-layer structural sandwich (Aicher and
Roth, 1987) and adopted by the Eurocode 5 – Part 1-1 (C.E.N. 1995). The elastic
solution for inelastic strains in the concrete slab and timber beam can be obtained by
solving the differential equation of the TCC. Such a solution is reported in Appendix A.
Let S be a generic effect (vertical displacement, slip, bending moment or axial
force) at the time t measured from the concrete casting, and t 0 the time of concrete
hardening (1 to 3 days). Let t G be the time when the dead load G is applied (usually
the time when the shores are removed) and t Q be the time when the live load Q is
applied. According to the superposition principle, it is possible to write:
14
S  S hG Q  S hs  S ely  S eld
(8)
where the superscripts denote the load conditions ( G : dead load; Q : live load; s :
concrete shrinkage; y : yearly inelastic strains due to thermo-hygrometric variations; d :
daily inelastic strains due to thermal variations) while the subscripts denote the type of
analysis to be employed ( h : hydro-viscoelastic; el : elastic).
The effects of the loads can be calculated as:
S hG Q  S hG  S hQ

S hi  Feli Eci ,eff , Ewi ,eff , k if ,eff
(9)

with i  G, Q
(10)
The effect of load i is signified by Feli and evaluated according to the approximate
elastic formulae reported in the Annex B of Eurocode 5 – Part 1-1 (C.E.N. 1995) by
substituting the effective moduli of concrete, timber and connection as described below:
Eci ,eff 
Ec t i 
1   c t , t i 
(11)
E wi ,eff 
Ew
1   w t , t i 
(12)
k if ,eff 
kf
1   f t , t i 
(13)
The creep coefficient of concrete  c t, t i  account for the creep phenomenon in such
material, whereas the creep coefficients of timber  w t, t i  and connection  f t, t i  have
to account for both creep and mechano-sorptive creep.
The effects of the concrete shrinkage are given by:
15

S hs  Felsh Ecs,eff , Ews ,eff , k sf ,eff

(14)
where Fel sh is evaluated according to the elastic formulae reported in Appendix A by
considering in them an inelastic strain  sh   c s t    c s t 0 ,  c s as being the
concrete shrinkage. The effective moduli Ecs,eff , Ews ,eff and k sf ,eff to be used are obtained
by Eqs. (11) to (13) by replacing E c t i  with E c t  and  k t, t i  with  k t ,t 0  , with
k  c, w, f . Since the concrete shrinkage increases in time with decreasing strain rate,
the Young modulus E c is still slight at the time of concrete hardening t 0 . Hence, the
quantity E c t  is evaluated at the time t of shore removal. This empirical criterion
provides satisfactory results since at that time (usually 7 to 14 days) the elastic modulus
of concrete is close to the medium value monitored during the entire service life.
The effects of the yearly and daily inelastic strains due to environmental
variations are given by:
), E w , k f

(15)
S eld  Feld Ec (t  ), Ew , k f

(16)
 y
S ely  Fel
E (t
c

 y
where t  is the final time of analysis, Fel
and Fel d are evaluated according to the
elastic formulae reported in Appendix A for an inelastic strain given, respectively, by:
 y   w u   w T   c T   w u u t   u t 0    w T T t   T t 0    c T T t   T t 0  (17)
 d   w T   c T   w T kTdaily   c T Tdaily
(18)
16
with u  ut  approximate history of average timber moisture content (Fig. 1c),
T  T t  approximate history of environmental temperature (Fig. 1a, thick solid line),
 dilatation coefficient for moisture content (subscript u ) or temperature (sub. T )
variation in timber (sub. w ) and concrete (sub. c ), Tdaily the average daily excursion
of environmental temperature during the year, and k a reduction factor to account for
the insulating properties of timber.
Careful consideration is required in order to define reasonable values for the
creep coefficients of the component materials. Several prediction models are available
for the concrete creep, such as the CEB 90 (1993) and the B3 (Bazant and Baweja 2000)
models. For timber and connection, the creep coefficients used in Eqs. (12) to (13) have
to account for both creep and mechano-sorptive creep. Both theoretical models (Ranta
Maunus 1975, Toratti 1992, Hanhijärvi and Hunt 1998) and approximate values (C.E.N.
1995) have been proposed for timber. In this paper, the CEB 90 prediction model has
been used for modeling creep and shrinkage of concrete. Two different possibilities
have been considered for modeling creep and mechano-sorptive creep of timber:
-
the Toratti model (1992). For a piecewise-linear moisture content history of
amplitude u and period t , the use of such a model leads to:
2 u
 cw
t    

E u   t   
  J w  E w u   1  e t
 w t ,    w c t     w ms t    w
 
 (19)
E w u ref   t d 


m
where  w c and  w ms are the creep and mechano-sorptive part of the global
coefficient  w , J w  0.7 / Ew u ref , u ref  0.20 , cw  2.5 , t d  29500 days and
m  0.21 (Toratti 1992, Fragiacomo and Ceccotti 2005);
17
-
the Eurocode 5 model (C.E.N. 1995), which provides the creep coefficients,
inclusive of mechano-sorptive effect, for different load durations, t   , and for
different service classes, i.e. for different values of maximum timber moisture
content and/or type of structure (indoor or outdoor). The values of creep coefficient
for the 3rd service class, which concerns structures exposed to the atmosphere, have
been fitted through the power-type formulation (first part of Eq. (19) – quantity
 w c ), obtaining the coefficients m  0.2193 and t d  708.7 days.
Since the dependency of the timber Young modulus on the moisture content is generally
negligible, Eq. (19) can be simplified by substituting E w u   E w u ref   E w , as it has
been done in Eq. (12). The creep coefficient of connection can be obtained, in absence
of experimental data, by doubling the creep coefficient of timber, according to prEN
1995-1-1 regulation (C.E.N. 2003):  f t ,   2 w t ,  .
Comparison with numerical solutions
Some numerical comparisons have been carried out in order to validate the proposed
simplified method. Figure 4 displays the trend in time at the 50th year of mid-span
vertical displacement, slip over the support, mid-span timber bending moment and axial
force for the Florence beam. Figure 5 displays the trend in time at the 50th year of the
mid-span vertical displacement for the EMPA beam. All figures concern the values
measured since the loading time t Q . Different curves are drawn (left ordinates):
a) the rigorous numerical solution (Eqs. (3) to (5)) obtained by using the model
proposed in the first part of the two companion papers (Fragiacomo and Ceccotti
2005) under the real environmental temperature and relative humidity histories, by
solving the diffusion problem over the timber cross-section for the Florence beam.
18
The real average moisture content history is assumed for the EMPA beam, since the
environmental relative humidity was not monitored;
b) the approximate numerical solution obtained by superimposing the elastic solution
for annual inelastic strains (Eqs. (3) to (5) with  c c  0 ,  c s  0 ,  w c  0 ,  w e ,u  0 ,
 w ms  0 , s f c  0 and s f ms  0 ) with the hydro-viscoelastic solutions for load and
concrete shrinkage (Eqs. (3) to (5) with  c T  0 ,  w u  0 and  wT  0 ). Each
solution is calculated by using the numerical model and by considering the
approximate piecewise-linear temperature and average moisture content histories
(thick solid lines in Figs. 1a and 1c) regarded as constant over the cross-sections;
c) only for the Florence beam, is the approximate numerical solution obtained by
superimposing the solution described in b) with the elastic one for inelastic strains
due to medium daily variations of temperature. This is equivalent to considering the
approximate temperature history given by the thick dash line in Fig. 1a instead of
the thick solid line when evaluating the effects of temperature variations;
d) the simplified algebraic solution, obtained by using the proposed procedure with the
timber and connection creep coefficient evaluated according to the Toratti model
(Eq. (19)), without daily variations of temperature;
e) same as d) but with creep coefficients evaluated according to the Eurocode 5 model.
Looking at the figures, the following remarks can be made:
-
For the Florence beam, the rigorous numerical solutions (curves “a”) fluctuate, with
reasonable approximation (Figs. 4a and 4b), in between the approximate numerical
solutions with (curves “c”) and without (curves “b”) daily variations of temperature.
This means that the simplified hypothesis of substituting the real timber moisture
content distribution with a piecewise-linear curve approximating the annual trend of
19
the average value (Fig. 1c) leads to satisfactory results in terms of deflection and
slip. This furnishes a confirmation, also, that the daily variations of moisture content
are negligible. Rigorous and approximate numerical solutions are nearly coincident
for the EMPA beam (Fig. 5), where no daily variation has been considered.
-
A somewhat larger difference can be noted for the bending moment and axial forces
(Figs. 4c and 4d). The approximation is, however, acceptable and on the safe side,
since the approximate numerical solutions fluctuate more than the rigorous ones.
-
The simplified algebraic solution fits in well with the approximate numerical one
when the same timber rheological model (Toratti 1992) is employed in both cases.
The Eurocode 5 model leads to solutions that go beyond those obtained using the
Toratti model; however the Eurocode 5 solutions are always conservative.
In Figure 6, the trend of mid-span deflection and slip are drawn for the Florence
beam during the entire service life. This figure considers the effects of both concrete
shrinkage and concrete shrinkage with applied live load. In observing the figure, the
following remarks can be made:
-
The approximate numerical solutions fit well with the rigorous numerical ones
during the entire service life and, above all, for the long term.
-
The simplified formulae allow for a good fit of the numerical solutions for both
concrete shrinkage and load. The use of different creep coefficients for timber and
connection has little effect on the simplified solutions due to concrete shrinkage.
However solutions due to load are affected by these coefficients, with the best
results being obtained by using the Toratti model.
-
The slip undergoes the largest error (about 16 %). The reason for this is the
approximation due to the variable connector spacing along the beam length, which
20
also has a similar effect on the elastic value due to live load (difference of 5 %). The
use of the effective spacing i f ,ef  is, in fact, only an empirical approximation in
order to extend the elastic formulae given in Appendix A and Annex B of Eurocode
5 to the case of TCC’s with variable connector spacing. The approximation is
generally rather good in terms of deflection, although less good in terms of slip.
The proposed method, for both simplicity and the acceptable approximation
attainable, is suitable for practical design. For the case of a simply supported beam, all
quantities to be checked in the long-term can be evaluated through analytical formulae,
without the need for numerical codes. The only problem arising is the evaluation of the
annual range of average moisture content for timber, u yearly . Such a quantity has to be
evaluated a priori according to the service class (indoor or outdoor structure) and the
average climate monitored in the reference region. This can be achieved by solving the
diffusion problem for different timber cross-sections or through experimental
measurements. However, the medium values of such quantity, along with the annual
and daily range of environmental temperature, T yearly and Tdaily respectively, could
be provided in specific regulations for different climate regions and service classes. The
creep coefficient for timber can be either provided by regulation, such as the Eurocode
5, in such a way as to account for both creep and mechano-sorptive creep, or
analytically evaluated according to a rheological model for timber, such as the Toratti.
Simplification for indoor structures and worked out example
The proposed approach allows an accurate evaluation of TCC’s in the long-term, which
is particularly important in outdoor conditions. For beams in heated indoor conditions
the environmental variations are lower. For example in the area of Florence it may be
21
assumed T yearly  10 C , T yearly  4 C and u yearly  0.02 , approximately one third,
half and half of the corresponding values for TCC’s in outdoor conditions, respectively.
When not overly accurate solutions are needed, the inelastic strains due to
environmental variations may be neglected and only the effects of dead, live load (Eqs.
(9) to (13)) and concrete shrinkage (Eq. (14)) are superimposed. However, it should be
kept in mind that this simplification leads to non-conservative solutions, thus its
application should be carefully considered.
As a practical application of the proposed simplified procedure, the timber
cross-sections of both Florence and EMPA beams have been redesigned in order to
satisfy the deflection limit of 1/200 of the beam length, which is generally required by
current codes of practice such as Eurocode 5 (C.E.N. 1995, 2003). The same
geometrical characteristics, loading and environmental histories as those reported in the
Section “Numerical Analysis” have been considered, with an average daily temperature
excursion of 8 °C for both beams. The Toratti theological model has been employed for
predicting creep and mechano-sorptive creep of timber and connection (Eq. (19)). Creep
and shrinkage of concrete have been evaluated according to the CEB 90 prediction
model (C.E.B. 1993). Solutions carried out using the proposed approach in terms of
minimum depth of the timber beam are 19.8 cm for the Florence beam and 14.3 cm for
the EMPA beam. The use of the classical approach (Ceccotti 1995), which neglects
effects of concrete shrinkage and inelastic strains due to environmental variations,
would lead to minimum depths of 17.4 cm and 12.2 cm, respectively. The classical
approach, hence, would underestimate the minimum timber depth of 12 % and 15 % for
the Florence and EMPA beam, respectively. Thus it is recommended that the proposed
approach be used for design of TCC’s, especially in the case of outdoor beams.
22
Conclusions
The second part of two companion papers investigates the importance of different
rheological phenomena, such as creep of concrete, creep and mechano-sorptive creep of
timber and connection, and the influence of inelastic strains produced by concrete
shrinkage and shrinkage/swelling due to thermo-hygrometric environmental variations,
on the behavior of Timber-Concrete Composite beams (TCC’s) in outdoor conditions.
Results are carried out using a numerical model that fully considers all the
aforementioned phenomena and leads to the rigorous solution by means of a step-bystep integration in time of the constitutive integral equations. The development of the
model and the validation against experimental results, including the influence of
environmental variations, are reported in the first part of two companion papers.
Three loading conditions act on the structure during the service life: dead and
live load, concrete shrinkage and inelastic strains due to thermo-hygrometric variations
of the environment. These conditions have been studied one by one in order to evaluate
their effects on the structural response. The following outcomes have been obtained:
-
The effect of concrete shrinkage, which is an increase in deflection and a decrease in
slip, is important. Creep and mechano-sorptive creep markedly increase deflection,
slip due to load and, as an absolute value, slip due to concrete shrinkage, although
deflection due to concrete shrinkage is decreased. The dependency of the timber
Young modulus on the moisture content is, instead, negligible.
-
The inelastic strains due to thermo-hygrometric environmental variations cause
daily and yearly fluctuations of deflection. Slip and internal forces also fluctuate and
may reach important values especially for beams subjected to outdoor conditions.
Creep and mechano-sorptive creep do not affect such solutions because of the cyclic
23
trend in time of environmental temperature and relative humidity histories. The
inelastic strains produce the maximum deflection when the concrete slab is cast at
the time of maximum temperature and minimum timber moisture content.
A simplified procedure for long-term evaluation of TCC’s is proposed, based on
the superposition of the hydro-viscoelastic solutions for load and concrete shrinkage
with the elastic solutions for inelastic strains due to annual and daily environmental
variations. Creep and mechano-sorptive creep are considered using the Effective
Modulus Method. The effects of loads are assessed by using the approximate elastic
formulae proposed by the Eurocode 5, while the effects of concrete shrinkage and
inelastic strains due to environmental variations are calculated by using the rigorous
elastic formulae. The trends in time of average timber moisture content and
environmental temperature are approximated by piecewise-linear curves. The accuracy
of the proposed procedure, especially in terms of deflection, which is the most
important quantity to be checked in the long-term, is highlighted by comparison with
numerical solutions. The choice of the creep coefficient for timber proposed by the
Eurocode 5 for structures in outdoor conditions leads to solutions going beyond the
numerical ones with respect to those obtained using the Toratti model, but always
remaining conservative. The comparison with the classical approach which considers
only effects of dead and live load shows an underestimation of the minimum timber
depth of about 15 %. The simplicity and accuracy attainable make the proposed
procedure suitable for practical design, particularly for TCC’s in outdoor conditions,
where the influence of rheological effect and thermo-hygrometric variations may be
relevant. The inelastic strains due to environmental variations may be disregarded for
approximate but non-conservative evaluations of beams in heated indoor conditions.
24
Appendix A
This Appendix reports the rigorous formulae for elastic analysis of simple supported
TCC’s with flexible connection subjected to inelastic strains uniformly distributed over
the concrete slab and timber beam:
v max  v max, full   v 
v max, full 
v  1
(20)
 n EI  full  EI abs l 2

 
EI  full
H
8
(21)


1
 1 

 cosh 0.5l 
(22)
8
l 
2
s f ( x)  s max, abs   s x 
s max, abs    n
 s x  
(23)
l
2
(24)
1
 tanh 0.5l   cosh x   sinh x 
0.5l
(25)
F f ( x)  k f s f  x 
(26)
N w ( x)   N c x   N w,max, full    x 
(27)
M i ( x)  M i ,max, full     x 
(28)
N w,max, full  
M i ,max, full 
with i  c, w
 n EI  full  EI abs EI abs


EI  full
H
H
 n EI  full  EI abs

 Ei I i
EI  full
H
with i  c, w 
(29)
(30)
  x   1  tanh 0.5l   sinh x   cosh x 
(31)
 n   w n   c n 
(32)
25
EI abs  Ec I c  Ew I w
(33)
EI  full  EI abs  EA*  H 2
(34)
EA*  Ec Ac Ew Aw
EI abs
(35)

EI  full
*
i f ,ef EA EI abs
kf

i f ,ef  0.75i f ,min  0.25i f ,max
(36)
(37)
where:
 n is the difference between the inelastic strains in timber  w n and concrete  c n ;
the subscripts abs and full refer to the case of TCC’s without connection and with
rigid connection, respectively;
the subscript max refer to the maximum value of an effect along the beam axis;
the subscripts c , w and f refer to concrete, timber and connection, respectively;
EI  is the flexural stiffness;
E , A and I are, respectively, the Young modulus, area and inertia moment of the
cross-section for the single component beam (concrete or timber);
k f is the connector stiffness and l is the beam length;
i f ,max , i f , min and i f ,ef are the maximum, minimum and effective spacing of connectors,
respectively;
H is the distance between the geometric centers of concrete and timber;
F f and s f are the connector shear force and relative slip, respectively;
N and M are the axial force and bending moment, respectively;
26
v and x are the deflection and the abscissa along the beam axis, respectively.
The coefficients  s 0 and   l / 2 , which are employed to evaluate the maximum
effects along the beam axis, assume the following simpler expressions:
 s 0  
tanh 0.5l 
0.5l
  l / 2  1 
1
cosh( 0.5l )
(38)
(39)
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Aug. 3, Saint John, New Brunswick, Canada, Vol. 2.
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Belgium.
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– Part 2: Bridges.” ENV 1995-2, Bruxelles, Belgium.
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– Part 1-1: General Rules and Rules for Buildings.” prEN 1995-1-1, Bruxelles,
Belgium.
Fragiacomo, M., and Ceccotti, A. (2005). “Long-term behavior of timber-concrete
composite beams. I: Finite element modeling and validation.” Journal of Structural
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Hanhijärvi, A., and Hunt, D. (1998). “Experimental indication of interaction between
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Kenel, A., and Meierhofer, U. (1998). “Long-term performance of timber-concrete
composite structural elements.” Report No. 115/39, EMPA, Dübendorf, Switzerland
(in German).
28
Kuhlmann, U., and Schänzlin, J. (2001). “Composite of vertically laminated timber
decks and concrete.” Proc., IABSE Conference “Innovative wooden structures and
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Mola, F., Creazza, G., and Pisani, M.A. (1989). “Static problems concerning the
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29
Figure Captions
Fig. 1. Maximum and minimum daily environmental temperature (a) and relative
humidity (b) monitored during the 5th year for the test performed in Florence, and
numerical average timber moisture content (c), with approximating piecewise-linear
histories
Fig. 2. Trend in time of mid-span deflection (a) and slip over the support (b) produced
by only concrete shrinkage (lower curves) and by both live load and concrete shrinkage
(upper curves) on the Florence beam for different constitutive models of the component
materials
Fig. 3. Trend in time of mid-span deflection produced by only concrete shrinkage
(lower curves) and by both live load and concrete shrinkage (upper curves) on the
EMPA beam for different constitutive models of the component materials
Fig. 4. Comparison between numerical and simplified approaches in terms of mid-span
deflection (a), slip over the support (b), bending moment (c) and axial force (d) in
timber produced by live load, concrete shrinkage and inelastic strains due to
environmental variations (left ordinate) and produced by only inelastic strains due to
environmental variations (right ordinate) for the Florence beam during the 50th year
Fig. 5. Comparison between numerical and simplified approaches in terms of mid-span
deflection produced by live load, concrete shrinkage and inelastic strains due to
environmental variations (left ordinate) and produced by only inelastic strains due to
environmental variations (right ordinate) for the EMPA beam during the 50th year
Fig. 6. Comparison between numerical and simplified approaches in terms of mid-span
deflection (a) and slip over the support (b) produced by only concrete shrinkage (lower
curves) and produced by both live load and concrete shrinkage (upper curves) for the
30
Florence beam
31
40
30 T [°C]
20
10
0
3/15/1995
-10
0
60
100
80
60
40
20 RH [%]
0
0
60
0.21
u [-]
0.20
0.19
0.18
0.17
3/15/1995
0.16
0
60
(a)
t [days]
8/2/1995 12/22/1995
120
180
240
300
(b)
120
180
360
t [days]
240
300
360
(c)
t [days]
8/2/1995 12/22/1995
120
180
240
300
360
Fig. 1. Maximum and minimum daily
environmental temperature (a) and relative
humidity (b) monitored during the 5th year for the
test performed in Florence, and numerical average
timber moisture content (c), with approximating
piecewise-linear histories
32
4
experimental
3
mech.-sorpt. of conn.
mech.-sorpt. of timber
c
d
creep of connection
creep of timber
dependency of Ew on u
f
g e
creep of timber
creep of concrete creep of conn.
d
g fb
mech.-sorpt. of conn.
creep of concrete
2
vel=0.88 mm
1
eac
t [years]
mech.-sorpt. of timber
0
0
10
20
30
40
50
4
sf/sf,el
3
sf,el=0.091 mm
a
b
experimental
2
mech.-sorpt. of conn.
mech.-sorpt. of timber
c
1
creep of connection
d
creep of concrete creep of timber
g e f
dependency of Ew on u
0
timb.of concr.
d b f e creep of conn. creep ofcreep
g
mech.-sorpt. of timber
mech.-sorpt. of conn.
c
a
-1
-2
t [years]
-3
0
10
20
30
40
50
Concr. shrink. Live load+Concr. shr.
a
b
v/vel
Concr. shrink. Live load+Concr. shr.
5
5
v/vel
experimental
4
a
b
c
3
d
2
e
f
g
mech.-sorpt. of timber
creep of connection
creep of timber
creep of concrete
dependency of Ew on u
creep of timber creep of concrete
degfb
creep of conn.
vel=2.78 mm
1
mech.-sorpt. of conn.
mech.-sorpt. of timber
mech.-sorpt. of conn.
0
0
10
20
30
c a
t [years]
40
50
Concr. shrink. Live load+Concr. shr.
(b)
(a)
Fig. 2. Trend in time of mid-span deflection (a) and slip over the support (b) produced by only concrete
shrinkage (lower curves) and by both live load and concrete shrinkage (upper curves) on the Florence beam
for different constitutive models of the component materials
Fig. 3. Trend in time of mid-span deflection
produced by only concrete shrinkage (lower
curves) and by both live load and concrete
shrinkage (upper curves) on the EMPA beam for
different constitutive models of the component
materials
33
0
d-simplified,Toratti
e-simplified,EC5
4
vel=0.88 mm
-1
t [years]
3
49
49.2
49.4
49.6
49.8
sf/sf,el
sf,el=0.091 mm sf/sf,el
5
2
4
1
3
0
a-rigorous numerical
b-approximate num.
c-approx.+daily var.
d-simplified,Toratti
e-simplified,EC5
2
1
0
50
49
49.2
49.4
(a)
a-rigorous numerical
b-approximate num.
c-approx.+daily var.
Total effect
2
1
1
0
0
d-simplified,Toratti
e-simplified,EC5
-1
49
49.2
49.4
49.6
-1
t [years]
49.8
2
2
Mw/Mw,el
49.8
-3
50
50
Nw/Nw,el
Nw,el=14.51 kN
Nw/Nw,el
0.5
1.5
Total effect
Mw,el=3.34 kNm
Mw/Mw,el
-2
t [years]
(b)
Effect of environm. variat.
3
49.6
-1
Effect of environm. variat.
1
5
6
2
0
1
a-rigorous numerical
b-approximate num.
c-approx.+daily var.
d-simplified,Toratti
e-simplified,EC5
0.5
0
-0.5
49
49.2
49.4
49.6
-0.5
-1
t [years]
49.8
Effect of environm. variat.
6
Total effect
v/vel
a-rigorous numerical
b-approximate num.
c-approx.+daily var.
Total effect
v/vel
Effect of environm. variat.
7
-1.5
50
(a)
(b)
Fig. 4. Comparison between numerical and simplified approaches in terms of mid-span deflection (a), slip
over the support (b), bending moment (c) and axial force (d) in timber produced by live load, concrete
shrinkage and inelastic strains due to environmental variations (left ordinate) and produced by only inelastic
strains due to environmental variations (right ordinate) for the Florence beam during the 50th year
34
v/vel
a-rigorous numerical
b-approximate num.
Total effect
5.5
v/vel
1.5
1
5
0.5
4.5
4
0
vel=2.78 mm
-0.5
d-simplified,Toratti
e-simplified,EC5
3.5
-1
t [years]
Effect of environm. variat.
6
3
49
49.2
49.4
49.6
49.8
50
Fig. 5. Comparison between numerical and
simplified approaches in terms of mid-span
deflection produced by live load, concrete shrinkage
and inelastic strains due to environmental variations
(left ordinate) and produced by only inelastic strains
due to environmental variations (right ordinate) for
the EMPA beam during the 50th year
6
vel=0.88 mm
v/vel
Load+Shrinkage
5
6
sf/sf,el
sf,el=0.091 mm
Load+Shrinkage
4
4
3
2
a-rigorous numerical
b-approximate num.
d-simplified,Toratti
e-simplified,EC5
2
a-rigorous numerical
b-approximate num.
d-simplified,Toratti
e-simplified,EC5
0
Shrinkage
Shrinkage
1
t [years]
0
0
10
20
30
40
50
-2
t [years]
-4
0
10
20
30
40
50
(a)
(b)
Fig. 6. Comparison between numerical and simplified approaches in terms of mid-span deflection (a) and
slip over the support (b) produced by only concrete shrinkage (lower curves) and produced by both live load
and concrete shrinkage (upper curves) for the Florence beam
35
Table 1. Geometrical, mechanical and rheological
properties of the Florence (Flor.) and EMPA beams
Beam
Flor. EMPA
Span length l [cm]
10x10-6 °C-1
Timber width [cm]
5x10-6 °C-1
Timber depth [cm]
3x10-3
Slab-timber gap [cm]
CEB 90 model
Concrete width [cm]
Toratti model
Concrete depth [cm]
Toratti model
Reinforcement [cm2]
2w Kenel
Dead load [N/m]
Toratti model
Shore removal [days]
15 cm 10 cm
Live load [N/m] [N]
45 cm 30 cm
Timber Young
Glued SFS
10000 12800 Connection type
modulus Ew [MPa]
rebars screws
Concr. strength [MPa] 30.4
34.8 Conn. stiffn. [N/mm] 25000 17000
Property
Beam
Property
Flor. EMPA
570
385 Concr. thermal coeff.
12.5
12 Wood thermal coeff.
50
18 Wood moisture coeff.
5
2
Creep, shr. of concr.
75
75 Creep of timber w
5
8
Mech-sorpt. of timb.
0.71
0.79 Creep of connection
1650 1700 Mech.-sorpt. of conn.
7
21 Min. conn. spacing
2000 2x6000 Max. conn. spacing
36
Table 2. Experimental-numerical comparison
during the 5th year of testing and influence of
different loading conditions on the maximum
numerical deflection and slip during the 50th year
Loading condition
Reference period:
5th year of testing
Elastic (live load)
Delayed (load+shr.)
Yearly inel. strains
TOTAL [0-5 years]
Reference period:
50th year (numerical)
Live load
Concrete shrinkage
Yearly inel. strains
Daily inel. strains
TOTAL [0-50 years]
Florence beam
(l=570 cm)
Deflection v
Slip sf
Num. Exp.
Num.
Exp.
[mm] [mm] [mm]
[mm]
0.88
0.87
0.091
0.045
2.20
2.19
0.071
0.147
0.83
0.39
0.111
0.134
3.91
3.45
0.273
0.326
v/l
v/vel
sf/l
sf /sf,el
[%]
[10-5] [%]
[10-5]
65.5
423
8.42
527
10.7
69
-3.30
-206
14.6
94
1.95
122
7.5
49
0.00
0
98.3
635
5.47
442
EMPA beam
(l=385 cm)
Deflection v
Num.
Exp.
[mm]
[mm]
2.78
2.66
6.73
7.73
0.47
0.49
9.98 10.88
v/l
v/vel
[%]
[10-5]
259.2
359
54.3
75
12.2
17
325.7
451
37
Table 3. Influence of different rheological
phenomena on the maximum deflection and slip
due to live load and concrete shrinkage during
the service life as percentage of the elastic value
due to live load
Rheological
phenomenon and type
of curve with reference
to Fig. 2 and 3
Elastic value
(g)
Ew = Ew(u)
(f)
Creep of concrete (e)
Creep of timber
(d)
Creep of connection (c)
Timb. mech.-sorpt. (b)
Conn. mech.-sorpt. (a)
TOTAL
Florence beam
EMPA beam
(vel=0.88 mm, sf,el=0.091 mm) (vel=2.78 mm)
v/vel [%]
sf /sf,el [%]
v/vel [%]
Load Shrink Load Shrink Load Shrink
100
104
100 -128
100
82
4
1
2
2
1
0
11
-17
-12
17
39
10
78
36
25
63
70
1
90
-42
208 -121
80
-14
68
18
42
43
45
0
72
-31
162
-82
24
-4
423
69
527 -206
359
75
38