Modelling of bending creep of low- and high-temperature-dried spruce timber

ORI GIN AL

Modelling of bending creep of low- and high-temperature-dried spruce timber

D. Honfi • A. Martensson • S. Thelandersson •

R. Kliger

Received: 11 February 2013 / Published online: 2 October 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract In the current project, a finite element model is developed to analyse the

long-term behaviour of timber beams. The time-dependent response of wood sub-

jected to bending and moisture changes is investigated in terms of strains and

stresses. A rheological model is implemented to capture the effects of creep,

mechano-sorption and hygroexpansion. The model is validated against test results

from Bengtsson and Kliger (Holzforschung 57:95–100, 2003). The results of the

analysis showed that the mechano-sorptive creep of low- (LT) and high-tempera-

ture-dried (HT) timber beams can be sufficiently modelled with a spring and a single

Kelvin body. The different mechano-sorptive behaviour of LT- and HT-dried

specimens is considered with different mechano-sorptive and shrinkage–swelling

parameters. The presented model could be used to derive general mechano-sorptive

parameters: (1) for better prediction of creep over the service life and (2) to provide

a basis of time-dependent probabilistic calculations for structural-sized timber in

serviceability limit state.

Introduction

The time-dependent behaviour of timber has a big influence on the deformations

(serviceability) and the stresses (ultimate limit state) of structures. The design codes

usually use creep coefficients to take into account the effects of the time-dependent

behaviour. However, creep coefficients consider the different aspects of creep (e.g.

load and moisture variation) in a very simplified way. To be more confident in

D. Honfi (&) � A. Martensson � S. Thelandersson

Division of Structural Engineering, Lund University, Box 118, 221 00 Lund, Sweden

e-mail: [email protected]

R. Kliger

Division of Structural Engineering, Chalmers University of Technology,

412 96 Gothenburg, Sweden

123

Wood Sci Technol (2014) 48:23–36

DOI 10.1007/s00226-013-0581-4

prediction of structural behaviour over the service life of the structure, calculation

methods should be based on rheological models based on laboratory test and data

from measurements on real structures (Schanzlin 2010).

Despite extensive research in the past decades, the long-term behaviour of wood

subjected to mechanical and moisture loading is not perfectly understood. However,

several constitutive models have been developed including the effect of creep,

mechano-sorption and hygroexpansion (Martensson 1988; Toratti 1992; Hanhijarvi

1995). These kinds of models are calibrated against some particular tests and can

hardly be generalised.

The general behaviour of timber is strongly dependent on the surrounding

conditions, such as (Morlier 1994):

• load and loading history;

• loading direction (in relation to the grain direction);

• time;

• moisture content and moisture history;

• moisture variations;

• temperature.

On the other hand, the material properties of wood have a strong influence on the

mechanical behaviour, since wood is a cylindrically orthotropic, viscoelastic

material with a high variability in strength and stiffness. These aforementioned

characteristics make it difficult to predict long-term deformations of wooden

structures.

Load-induced deformation is usually separated into two parts, such as elastic and

delayed deformation, i.e. normal creep. Elastic deformation is defined as the

immediate deformation when applying or removing load. However, it is often

difficult to make this separation, since the loading itself takes some time.

The viscoelastic creep is the gradually increasing deformation—with a decreas-

ing rate—after the immediate elastic response when the load is kept unchanged. If

the load is removed, recovery takes place, which can also be separated into two

parts: an immediate elastic recovery and a delayed viscoelastic recovery with a

slowing rate.

As a hygroscopic material, wood binds water from moisture in the surrounding

climate. The moisture content in the timber affects several of the material properties

such as strength, modulus of elasticity and creep. For moisture contents up to the

fibre saturation point, the modulus of elasticity decreases as the moisture content

increases. The reduction in the modulus of elasticity softens the wood and thus

increases the elastic deformations and the normal creep.

Mechano-sorptive creep is the phenomenon caused by the moisture content

change in timber under mechanical stress. The mechano-sorptive creep can be

defined as the additional creep due to moisture variation, compared to normal creep

obtained under constant moisture content given the same conditions otherwise.

Hygroexpansion (also called shrinkage and swelling) is the dimensional change

in wood due to changes in moisture content. These dimensional changes take place

in all three directions, but the magnitude in longitudinal direction is smaller

24 Wood Sci Technol (2014) 48:23–36

123

compared to the transversal ones. The shrinkage and swelling are usually considered

to be linear with moisture content changes below the fibre saturation point.

In the current study, an attempt is made to simulate the tests made by Bengtsson

and Kliger (2003).

Tests

In the research project from Bengtsson and Kliger (2003), 15 butt logs from Norway

spruce (Picea abies) were used. From each log, three battens were sawn, and from

the centre batten, six studs, 45 9 70 9 5,000 mm3, were cut. The sawing pattern is

shown in Fig. 1. The studs numbers 1, 2 and 3 were low-temperature-kiln-dried—

conventional method—while the studs numbers 4, 5 and 6 were high-temperature-

dried. The LT-drying was performed at 70 �C during 168 h, while the HT-drying

was performed at a maximum temperature of 115 �C for approximately 30 h.

The creep test specimens were conditioned until equilibrium moisture content at

20 �C and 65 % relative humidity (RH) before the creep tests started. The ends of

the specimens were sealed by silicone to avoid moisture transport in the longitudinal

direction. Six specimens from one tree were loaded at the same time, see Fig. 2. The

dimensions of the studied beams were 45 9 70 9 1,100 mm3. They were loaded in

bending for 240 days in RH varying between 30 % and 90 % and at a constant

temperature of 20 �C. The specimens were loaded flatwise in four-point bending to

a maximum bending stress of 10 MPa. The duration of the moisture cycles was

2 9 14 days (drying and wetting).

Methodology

Constitutive models

The constitutive equation of wood when subjected to mechanical loading and

moisture variation can be given as:

eðtÞ ¼ eviscðtÞ þ emsðtÞ þ euðtÞ ð1Þ

where e is the total strain, evisc is the viscoelastic strain, ems is the mechano-sorptive

strain and eu is the shrinkage–swelling strain due to hygroexpansion.

5 m

root

top

intermediateouter

core

studs45 x 70 x 5000 mm

290 mm5000

123

456

Distortion test specimen

sticks

Creep test specimen

Fig. 1 Cutting of specimens from the logs (Bengtsson and Kliger 2003)

Wood Sci Technol (2014) 48:23–36 25

123

Viscoelastic creep

The viscoelastic behaviour of timber can be described by the viscoelastic strain rate

_evisc, which can be further divided into an elastic part _eel and a viscous part (normal

creep) _ec:

_evisc ¼ _eel þ _ec ð2ÞThe elastic strain rate _eel can be expressed as:

_eel ¼ J0ðuÞ _r ð3Þ

where J0 is the elastic compliance depending on the moisture content (by weight) u,

and _r is the stress rate.

The normal creep can be modelled with m Kelvin elements in series, and thus, the

normal creep strain rate _ec is given as:

_ec ¼Xm

i¼1

Jir� ec;i

si

� �ð4Þ

where Ji is the creep compliance, si is the relaxation time of the ith Kelvin element

and ec;i the normal creep strain caused by the ith Kelvin element.

Mechano-sorptive creep

There are several models to describe the mechano-sorption, such as constant slope

models (Ranta-Maunus 1975), the creep limit models (Hunt and Shelton 1988), two

slope models (Leicester 1971; Martensson 1988) and combined models (Toratti

L = 1040a

400

500a

45

Fig. 2 Load arrangement for the creep tests in bending (Bengtsson and Kliger 2003)


123

1992; Svensson and Toratti 2002). In the current study, the creep limit model is

used. The mechano-sorptive strain rate is divided into two parts:

_ems ¼ _emsv þ _emsr ð5Þ

where _emsv and _emsr are the recoverable (variable) and the irrecoverable (residual)

parts of the mechano-sorptive strain rate, respectively. The residual part of the

mechano-sorption is modelled by n Kelvin elements in series:

_emsr ¼Xn

j¼1

Jmsr;jr� emsr;j

smsr;j

� �_uj j ð6Þ

where Jmsr;j is the compliance, smsr;j is the relaxation parameter of the jth Kelvin

element, emsr;j the residual part of the mechano-sorptive creep strain caused by the

jth Kelvin element and _u is the rate of the moisture content change.

The recoverable part of the mechano-sorptive strain rate is proportional to the

moisture content rate _u and the total strain e:

_emsv ¼ �be _u ð7Þ

where b is a material parameter.

Shrinkage and swelling

The free shrinkage rate _eu is defined as a linear function of moisture content rate _u.

_eu ¼ a _u ð8Þ

where a is the shrinkage–swelling coefficient, which is assumed to be independent

of moisture.

Moisture transport

The moisture transport in wood can be described by similar equations to heat

transfer in solids. The diffusion within the material is expressed by Fick’s second

law:

ow

ot¼ r Dwrwð Þ ð9Þ

where w is the moisture content per volume, and Dw is the diffusion coefficient.

At the surface, the exchange of moisture between air and wood is modelled as a

surface flux q according to Fick’s first law.

q ¼ bw wsurf � weq

� �ð10Þ

where bw is the mass transport coefficient (or surface emissivity) with respect to

moisture content, and wsurf and weq are the moisture content at the surface and in the

air (equilibrium moisture content), respectively.


123

Modelling

The FEM model

A finite element model has been developed to study the effect of moisture cycling in

timber. The FEM modelling was carried out using COMSOL Multiphysics 4.3a

(COMSOL AB 2012).

Two models are developed and coupled together. Model #1 is intended to

calculate the moisture distribution within a cross-section of the beam in both

directions, while Model #2 determines the strains, stresses and displacements of the

whole beam. The coupling ensures that the distribution of moisture content and

moisture content rate from Model #1 to Model #2 can be obtained.

In Model #1, the Coefficient Form Partial Differential Equation (PDE) Interface

is used to implement the moisture balance equations in COMSOL. The PDE

interface contains a linear equation where both sides are sums of a known function,

the unknown functions and their partial derivatives, multiplied by known

coefficients. The dependent variable—in this model the moisture content w—is

an unknown function on the computational domain, i.e. the cross-section of the

beam.

In Model #2, the Solid Mechanics Interface and 4 Distributed Ordinary

Differential Equation (DODE) Interfaces are used to couple mechanical stress and

strain using the constitutive relations described in the previous sections.

The Solid Mechanics interface uses an orthotropic, linear material model to

calculate the stresses (r), strains (e) and the displacement field (d) on the predefined

2D domain.

A DODE Interface contains equations involving functions and their derivatives.

The derivatives are with respect to one independent variable only. In the current

model, the DODE interfaces represent the constitutive equations for creep,

mechano-sorption (reversible and irreversible part) and hygroexpansion. The

dependent variables are the respective strains.

Model #2 is then coupled to Model #1 to allow the numerical simulation of the

moisture distribution, as well as the resulting mechanical stresses and strains in

wood (see Fig. 3).

The temperature is assumed to be constant in the current study.

Model #1

The first model contains a 2D domain using quadratic Lagrange elements. The

moisture transport was modelled in both—horizontal and vertical—directions

within the cross-section. The geometry and the mesh are shown in Fig. 4.

The initial relative humidity is set to 65 % and then changed to 30 % at the

boundaries. Both at this initial step and at all subsequent moisture cycles, RH was

changed gradually to avoid numerical problems, see Fig. 5.

The coefficients of the PDE are the diffusion coefficient and the moisture

capacity. The first one is a function of the moisture content (Fig. 6a), and thus, the

moisture isotherm of wood (Fig. 6b) is required. The latter one in this description


123

owot¼ r Dwrwð Þ

� �(9) is equal to 1. As a simplification, the same isotherm is used

for adsorption and desorption. The mass transport coefficient to calculate surface

flux is constant bw = 1.3 9 10-7 m/s (Toratti 1992).

Model #2

In the second model, a 2D domain is defined with quadratic Lagrange elements. The

geometry and the mesh with the applied load and the boundary conditions are

Fig. 3 Coupling of the models and interfaces used in COMSOL

Fig. 4 Geometry and FE-mesh for model 1


123

presented in Fig. 7. One half of the beam is modelled applying symmetry in the

middle. The support at the other end is modelled as a roller.

To avoid local stress concentrations at the load (875 kN) and the point support,

the concentrated forces were distributed to rigid surfaces with a length of 20 mm.

Linear elastic material model

In the Solid Mechanics interface, the plane stress condition is assumed, and a linear

elastic orthotropic material model was defined. The parameters of the linear elastic

model are presented in Table 1. For both the LT- and HT-dried wood, the 6

specimens are chosen for modelling, cf. Fig. 1. Specimens 1, 2 and 3 were LT-

dried, while the specimens 4, 5 and 6 were HT-dried. The density q and the

modulus of elasticity E0 are taken from the measurements on the specimens being

modelled.

The modulus of elasticity was assumed to be a function of the moisture content

(with slight modification of the formula given by Toratti 1992):

Fig. 5 Modelling of RH changes with smooth steps

Fig. 6 a Diffusion coefficient and b Sorption isotherm of wood


123

EðuÞ ¼ E0

ð1� 1:06uÞð1� 1:06u0Þ

ð11Þ

where E0 is the Young’s modulus at a reference moisture content u0.

Normal creep model

The parameters from the Kelvin elements for normal creep are taken from Toratti

(1992). Six Kelvin bodies are used in series with the parameters given in Table 2.

These values are just accepted and not calibrated against the test results, since the

effect of normal creep is less significant than that of the mechano-sorptive creep. It

is assumed in the current study that the normal creep is the same for LT- and HT-

dried wood.

Mechano-sorptive creep model

In the current modelling task, the mechano-sorptive creep has the largest influence

on the results, and thus, the main goal is to find reasonable parameters for the

constitutive model of mechano-sorption for both LT- and HT-dried specimens.

Fig. 7 Geometry, load and mesh of Model #2

Table 1 Mechanical

parameters of Model #2Description Density Reference Young’s

modulus in the

longitudinal direction

Reference

Young’s modulus

in the transversal

direction

Name and

unit

q (kg/m3) El,0 (MPa) Et,0 (MPa)

LT

1 449 13,260 600

2 427 12,010 600

3 404 8,920 600

HT

4 400 9,850 600

5 440 12,890 600

6 497 15,260 600

Reference Bengtsson and Kliger (2003) Assumed value


123

For sake of simplicity, only one Kelvin element is used to model the residual

mechano-sorptive creep. The parameters of mechano-sorption are given in Table 3.

These parameters were determined to provide good agreement with the test results.

Shrinkage–swelling model

The shrinkage–swelling coefficient is determined from tests made by Bengtsson and

Kliger (2003). From each stud, 15 sticks (11 9 11 9 200 mm3) were sawn, see

Fig. 1. The strain and absorbed water during a single moisture cycle from

RH = 90 % to RH = 30 % were measured. From the results, the shrinkage–

swelling coefficient can be calculated (Table 4).

Coupling of the models

The two previously described models were coupled using Linear Projection and

Linear Extrusion Coupling options in COMSOL. First, the moisture content

distribution of the two-dimensional cross-section in Model #1 was projected to its

vertical centreline, i.e. the average moisture content along the centreline was taken.

Then, this average moisture content distribution was assigned to the points of the 2D

domain in Model #2. Thus, the moisture contents in Model #2 represent the average

moisture content along the vertical centreline of the cross-section.

Results

The results of the test from Bengtsson and Kliger (2003) showed that the mechano-

sorptive effect is higher at LT-dried specimens than that for HT-dried ones. The

relative creep of LT-dried beams after 8.5 moisture cycles was about 2.0–2.3, while

the same value for HT-dried beams was between 1.4 and 1.7. Figure 8 shows how

the simulations fit the test results. The shape of the curves is mostly influenced by

the mechano-sorptive parameters. Parameter Jmsr determines the value of the final

creep and smsr controls how fast it will approach the final value, whereas b defines

the changes in creep within the moisture cycles. The overall behaviour can be

captured with the model very well; however, the increased deflection at the second

moisture cycle is usually underestimated by the model. A better fit at the earlier

Table 2 Creep parameters for

Model #2Ji (-) si (d)

0.0686 0.01

-0.0056 0.1

0.0716 1

0.0404 10

0.2073 100

0.5503 5,000


123

cycles could probably be achieved by applying multiple mechano-sorptive Kelvin

elements.

It should also be mentioned that some parameters, for which no exact information

is available of, can affect the relative creep of the specimens. These are, for

example, the rate of relative humidity change between the cycles, mass transfer

coefficient and the sorption isotherm of the actual wood material. The difference in

the curvature of the test and the simulated graphs within a single moisture cycle

could be explained by those.

Table 3 Parameters for the mechano-sorptive creep

Specimen Drying method Jmsr (1/MPa) smsr (-) b (-)

1 LT 0.52J0 0.4 1.3

2 LT 0.65J0 0.4 1.6

3 LT 0.74J0 0.4 1.6

4 HT 0.22J0 0.35 1.3

5 HT 0J0 0.35 0.7

6 HT 0.35J0 0.35 1.3

Table 4 Parameters for the

shrinkage and swellingSpecimen Drying method a (-)

1 LT 5.56 9 10-3

2 LT 7.95 9 10-3

3 LT 8.99 9 10-3

4 HT 11.59 9 10-3

5 HT 11.42 9 10-3

6 HT 6.10 9 10-3

1.0

1.5

2.0

2.5

0 50 100 150 200 250

Time [d]

3(LT)

Rel

ativ

e cr

eep

[-] 2(LT)

6(HT)

4(HT)

5(HT)

1(LT)

Fig. 8 Relative creep of the LT- and HT-specimens–test (dashed lines) versus simulation (cont. lines)


123

Figure 9 illustrates the development of different strain components at the lower

edge of the midsection for specimen 6 (HT-dried) as an example. Each component

may vary depending on the numerical parameters of the model (e.g. mesh,

convergence criteria, time step size etc.); however, it will not affect—or only

marginally—the total strain and thus the deflections of the beam. Since the authors

are interested in the general behaviour, it is acceptable and allows to use less time-

consuming models. Furthermore, the definition of the different strain components is

much more a theoretical consideration than a real physical feature.

The changing of the average moisture content along the centreline of the cross-

section at different depths is presented in Fig. 10. H represents the surface and

0.5H the middle of the cross-section.

Conclusion

The results show that the mechano-sorptive creep of LT-and HT-dried timber beams

can be well modelled using the creep limit model with a spring (variable part) and a

-0.05%

0.00%

0.05%

0.10%

0.15%

0 50 100 150 200 250

Stra

in [

-]

Time [d]

εelεcεmsεuεtot

Fig. 9 Development of the strain components at the lower edge of the midsection (specimen 6)

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0 50 100 150 200 250

H

0.9H

0.8H

0.7H

0.6H

0.5H

Time [d]

u [-

]

Fig. 10 Changing of the average moisture content along the centreline of the cross-section (specimen 6)


123

single Kelvin body (residual part) in the COMSOL Multiphysics environment. The

fit with the test results is somewhat less accurate for the moisture cycles at the

beginning. However, that is less of a problem, since one is usually interested in the

final value of creep.

The different mechano-sorptive behaviour of the specimens cut from the same

wood, but treated differently, could be explained by the changed mechano-sorptive

and shrinkage–swelling parameters. The main difference comes from the decreased

Jmsr compliance of the mechano-sorptive dashpot of the HT-dried specimens. When

developing even more simplified models for designing timber structures, this

information about the effect of drying should be considered.

To achieve better agreement with the results, application of more Kelvin

elements or a more advanced model is recommended, especially when it comes to

modelling early stages of mechano-sorption. An advanced model could include

‘‘memory effects’’ of wood, which means that its mechano-sorptive behaviour is

different for moisture levels attained earlier during moisture cycling (Martensson

1992) and/or application of multiple mechano-sorptive Kelvin elements (Svensson

and Toratti 2002). The more sophisticated models can be used in order to discover

what phenomena that are of importance in the behaviour of wood subjected to

humidity changes. However, in order to find out more general knowledge, less

complex models are preferred; since the more parameters that are included in the

model, the harder it is to find the parameters in a well-defined manner.

In the future, the presented sufficiently accurate yet relatively simple model could

be used to derive general mechano-sorptive parameters for timber. This could be

used to give better estimation of the creep at variable loading and climate

conditions. Furthermore, time-dependent probabilistic calculations could be carried

out to predict the probability of serviceability failure for structural-sized timber.

References

Bengtsson C, Kliger R (2003) Bending creep of high temperature dried spruce timber. Holzforschung

57:95–100

COMSOL AB (2012) COMSOL Multiphysics user’s guide, version 4.3a

Hanhijarvi A (1995) Modelling of creep deformation mechanisms in wood, Technical Research Centre of

Finland Espoo, Finland, VTT Publications 231

Hunt D, Shelton C (1988) Longitudinal moisture-shrinkage coefficients of softwood at the mechano-

sorptive creep limit. Wood Sci Technol 22:199–210

Leicester RH (1971) A rheological model for mechano-sorptive deflections of beams. Wood Sci Technol

5:211–220

Martensson A (1988) Tensile behavior of hardboard under combined mechanical and moisture loading.

Wood Sci Technol 22:129–142

Martensson A (1992) Mechanical behaviour of wood exposed to humidity variation, PhD Thesis, Lund

University, Sweden. Report TVBK-1006

Morlier P (1994) Creep in timber structures. RILEM, Report 8

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9:189–205

Schanzlin J (2010) Modeling the long-term behavior of structural timber for typical service class-II-

conditions in South-West Germany, Habilitation, University of Stuttgart, Germany


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Svensson S, Toratti T (2002) Mechanical response of wood perpendicular to grain when subjected to

changes of humidity. Wood Sci Technol 36(2):145–156

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Documents

Modelling of bending creep of low- and high-temperature-dried spruce timber