Analysis of Crack Propagation due to Thermal Stress in

Journal of Advanced Concrete Technology Vol. 5, No. 1, 99-112, 2007, February 2007 / Copyright © 2007 Japan Concrete Institute 99

Scientific paper

Analysis of Crack Propagation due to Thermal Stress in Concrete Considering Solidified Constitutive Model Worapong Srisoros1, Hikaru Nakamura2, Minoru Kunieda3 and Yasuaki Ishikawa4

Received 30 April 2006, accepted 16 December 2006

Abstract In recent years, interest in early age concrete cracking has increased due to its effects on the durability and performance of concrete structures. A time-dependent material model and a structural analysis method have been developed to evaluate thermal cracking behavior. To simulate such behavior at early ages, a solidified constitutive model is proposed, which is based on the solidification concept with dependence on time and strain histories. The unified numerical model consists of a Rigid-Body-Spring Network, representing the structural behavior, combined with a truss model to represent heat transfer. Wall concrete structures are analyzed to verify the solidified constitutive model and the overall approach. The proposed model results and the experimental results show reasonable agreement in terms of cracking behavior, stress distributions and structural deformations.

1. Introduction

Thermal cracking in concrete, which is caused by changes in concrete volume during the hydration process, is a major cause of initial defects and structural per-formance deterioration during the service life of concrete structures. Thermal stresses are the results of internal strains under the internal and external restraints within the structure due to temperature changes during the hardening process or changes in the ambient temperature conditions. In addition, initial cracks may accelerate the corrosion in the concrete structure caused by the ingress of chloride ions to the reinforced steel members passing through these cracks. Therefore, in order to control and limit these undesirable effects on the durability and performance of concrete structures, the crack patterns and cracking behavior due to thermal stress of early age concrete have to be evaluated accurately at the design stage. The results of the analysis help engineers modify the constructional procedures, concrete mix design and structural details to ensure proper confinement of con-crete structures under the environmental conditions at a particular site.

To predict the thermal stresses during the hydration process in concrete structures, knowledge of the proper-ties of early age concrete is strongly required. While many researchers have developed constitutive models

for early age concrete before concrete cracking, few researchers have considered also the behavior after cracking. For instance, the elastic incremental method is one of the early age concrete models used to evaluate the behavior of concrete before cracking. Ishikawa et al. (1997) also investigated wall concrete structures by using the FEM Method taking the creep function into consid-eration, but he was not able to obtain the structural be-havior after cracking. Since no time-dependent constitu-tive model of concrete before and after cracking has been fully elucidated, the solidified constitutive model is proposed in this paper as a material model for repre-senting the stress strain relationship both before and after cracking for early age concrete. A combination of mate-rial and structural analysis models is required to evalu-ate thermal cracking behavior and thus the objective of this paper is to develop time-dependent constitutive and structural analysis models to evaluate structural concrete behavior due to thermal stresses, both before and after cracking.

For the structural analysis model, the unified numeri-cal model (Nakamura et al. 2006) consisting of the Rigid-Body-Spring Networks (RBSN) model (Kawai 1978) as the basis for a structural model and a truss network model as a basis for a mass transfer model (Bolander and Berton 2004) was used because it can clearly display the initial cracks and crack propagations without having to define the position of cracks before analysis. Moreover, random geometry was introduced using Voronoi diagrams in order to reduce mesh bias on crack propagation (Bolander and Saito 1998). At this stage in the development of this proposed method for evaluating the thermal cracking behavior of concrete, the important effects of creep, maturity, etc., were not given primary consideration. To verify the proposed material and structural models, thermal stress and ther-mal cracking in wall concrete structures were later evaluated and compared with the experimental results.

1Doctoral Engineer, Dept. of Civil Engineering, Nagoya University, Japan. E-mail: [email protected] 2Professor, Dept. of Civil Engineering, Nagoya University, Japan. 3Associate Prof., Dept. of Civil Engineering, Nagoya University, Japan. 4Associate Prof., Dept. of Civil Engineering, Meijo University, Japan.

100 W. Srisoros, H. Nakamura, M. Kunieda amd Y. Ishikawa / Journal of Advanced Concrete Technology Vol. 5, No. 1, 99-112, 2007

2. Structural and heat transfer model

2.1 Concrete structural model A discrete model has been applied for cracking in con-crete structures because such a model shows the discon-tinuous displacement field. The Rigid-Body-Spring Network is one of the discrete models used for analyzing fracture behaviors of concrete besides the Finite Element Method (FEM).

According to RBSN, concrete is modeled as an as-semblage of rigid particle elements interconnected by zero-size springs distributing the energy along their boundaries, in which each rigid particle has two transla-tions and one rotational degree of freedom defined at the nuclei within. The interface between two particles con-sists of three individual springs namely normal, tangen-tial, and rotations, as shown in Fig. 1, in which kn, kt and kθ denote the stiffness of the normal, shear (tangential), and rotational springs. These stiffnesses are simple functions of the distance, hIJ, between the nuclei of par-ticles I and J, and the length of the common Voronoi edge, sIJ.

12/

//

2IJn

IJIJt

IJIJn

skk

hEAkhEAk

=

==

θ

(1)

where AIJ = sIJt, t = the thickness of the planar model and E = the elastic modulus of the continuum material. The response of the spring model yields an understanding of the interaction between particles instead of the internal behavior of each particle based on continuum mechanics. Since concrete cracks initiate and propagate along in-terparticle boundaries, the crack pattern is strongly af-fected by the mesh design. Therefore, random geometry using Voronoi diagrams is applied to partition the mate-rial onto an assembly of rigid particles (Bolander and Saito 1998). The random geometry of the networks does not represent any structural feature within the concrete material, but rather is used to reduce mesh bias on po-tential crack directions.

2.2 Heat transfer model Heat transfer is a continuous flow and it is usually ana-lyzed by using the continuum model with the Finite Element approach, which is represented by a partial differential equation. However, RBSN is used for struc-tural analysis, which does not require continuality. Therefore, the truss network model is applied for heat transfer analysis. That is, truss elements link each of the Voronoi nuclei and the intermediate points of the Voronoi facet to form networks, as shown in Fig. 2. Thus, a sim-plified one-dimensional diffusion equation using truss elements is employed to carry out heat transfer. 2.3 Unification of structural model and heat transfer model To evaluate concrete structural behaviors due to heat

transfer within bulk concrete, the structural model and heat transfer model should be combined together. Therefore, the combination of the structural model and heat transfer model is proposed, whichin initial strain problems obtained from the heat transfer model are solved in the structural model. Figure 3 shows the uni-fied numerical model comprising the structural model and heat transfer model.

3. Heat transfer and thermal stress analysis

3.1 Heat transfer in truss elements A truss element is used to model one-dimensional heat transfer, where the heat flow is assumed to be propor-tional to the temperature gradient between the nodes at both ends of the truss element. Each element has only one degree of freedom at each node as temperature. Based on the governing equation of the heat transfer on one dimension, therefore, the discrete finite element for

Voronoi diagram

kn

kt

hIJ

kθ

I J

Voronoi diagram

kn

kt

hIJ

kθ

I J

Fig. 1 Rigid-Body-Spring Networks (RBSN).

NodeTrussNodeTruss

Fig. 2 Truss network model.

Unified numerical model

Structural model Mass transfer model(RBSN) (Truss network model)

Unified numerical model

Structural model Mass transfer model(RBSN) (Truss network model)

Fig. 3 Unified numerical model.

W. Srisoros, H. Nakamura, M. Kunieda amd Y. Ishikawa / Journal of Advanced Concrete Technology Vol. 5, No. 1, 99-112, 2007 101

heat transfer analysis has the following finite form,

[ ] { } [ ]{ } { }FTKtTC =+∂∂ (2)

Heat conduction matrix,

[ ] [ ] [ ] VNNccV

T d∫= ρ (2a)

Heat capacity matrix,

[ ] [ ] [ ][ ] [ ] [ ] SNNhVBDBkT

S

T

Vdd

2∫∫ += (2b)

Load vector,

[ ] [ ] SNhTfT

Sd∫ ∞−= (2c)

Shape function,

[ ] [ ]

−==

Lx

LxNNN ji 1

Material property matrix,

[ ] [ ]xxKD =

[ ]

−=

LLB 11

where {T} = temperature (°C), ρ = density of concrete (kg/m3), c = specific heat (J/kg⋅°C), h = heat transfer coefficient (J/m2⋅s⋅°C) and T∞ = atmosphere temperature (°C). [N] is the shape function of a one-dimensional element defined by the local coordinate system and L is the element length (m). Thus, for each element, Equation (3) represents heat transfer in finite form as follows.

=

−+

∂∂

∂∂

+

−−

∞ 000

2112

61

1111

2

2

1

2

1

TThA

tT

tT

ALTT

LAk

ω (3)

where T1, T2 = temperature at both nodes of the truss element, A = truss element area and k = thermal conduc-tivity (J/m⋅s⋅°C). In this study, ω in the second term is added into Equation (3). The parameter ω is added to convert the volume of a Voronoi polygon to the volume of truss elements due to the overlap volume of adjacent truss elements (Bolander and Berton 2004). The Crank-Nicolson scheme is used for direct integration of the heat transfer equation. 3.2 Internal stain and stress due to temperature change The development of thermal strain, εT, can be calculated by the simplified scalar expression presented in Equation

(4). The magnitude of the thermal strain is directly pro-portional to the magnitude of the temperature change, ∆T, and it is the function of the coefficient of linear thermal expansion, αT (strain/°C), as well.

TT TstrainThermal αε ⋅∆=, (4)

The internal strains of each truss element obtained from heat transfer analysis are introduced into the normal springs at the same position of RBSN as structural analysis. Thus, thermal stresses can be calculated under the internal and external restraints taking into considera-tion concrete properties.

3.3 Verification of heat transfer and thermal stress analysis in two-dimension A commercial program, JCMAC1 (JCMAC1, 2003) developed from the FEM method, was used for analysis, comparing its results with those of the unified numerical model in which the elastic incremental method was as-sumed to obtain stress distributions. Figure 4 displays the configuration and dimension of a rectangular con-crete specimen with 200 rectangular elements for FEM method analysis and 300 Voronoi polygons for the uni-fied numerical model. The black line and red dashed line represent an adiabatic boundary and heat transfer boundary, respectively. Table 1 lists the properties of the concrete and the testing conditions.

The adiabatic temperature rise and Young’s modulus were assumed as Equations (5) and (6), respectively. E28 is Young’s modulus at 28 days and set to 28000 MPa.

)(tν is a volume function, as shown in Equation (7), which define the changes of concrete properties as de-scribed in section 4.

( )teT 8.0145 −−×= (5)

28( ) ( )= ⋅E t E v t (6)

Adiabatic boundary

2000 mm

200

mm

Point C

Point B

Point A

Point D

2000 mm

200

mm

A

B 300 elements

Heat transfer boundary

Point C

Point B

Point A

Point D

Adiabatic boundary

2000 mm

200

mm

Point C

Point B

Point A

Point D

2000 mm

200

mm

A

B 300 elements


2000 mm

200

mm

A

B 300 elements


Point C

Point B

Point A

Point D

Fig. 4 Two-dimensional models for heat transfer simula-tion.


( ) ( )30

28 2⋅

=⋅ +

tv tt

(7)

where T = temperature and t = time (day). Figures 5 and 6 show the comparison results between the unified nu-merical model and FEM model for points A, B, C and D. As shown in Fig. 5, the results from the truss network model correspond very well with the FEM method. Moreover, the horizontal stress distributions, as shown in Fig. 6, are also similar to those of the FEM method. Therefore, it can be concluded that the method combin-ing RBSN and the truss network model can be applied to analyze thermal stress behaviors in early age concrete similarly to the FEM model. 4. Concrete material models

The accuracy of thermal stress analysis depends primar-ily on how the required mechanical properties are de-scribed both for early age and hardened concrete. For the hardened concrete, the properties and strength of the concrete are assumed to be constant and thus independ-ent of time. On the other hand, the mechanical properties of early age concrete rapidly change, particularly in the period when the concrete changes from a liquid to a solid. The key mechanical properties required for analysis at an early age are the modulus of elasticity, the tensile strength, and the properties governing the viscoelastic behavior of the material (Mihashi and Leite 2004).

The lack of the time-dependent mechanical properties of concrete especially after cracking is the main problem in the study of early age concrete structural behaviors. Therefore a time-dependent constitutive model of early age concrete before and after cracking should be pro-posed. In this paper, a solidified constitutive model is presented based on the solidification concept, which is different from the concept presented by Bazant et al. (1989). Since the solidification theory proposed by Ba-zant is based only on the equilibrium condition of force for the total volume considering static constraints, the stress increment of each solidified element cannot be determined uniquely. On the other hand, the solidified constitutive model is based on kinematic constraints whereby concrete elements solidified at different ages are subjected to the same strain increments. Therefore, it is possible to consider the nonlinear stress-strain relation easily and calculate the stress increment of each solidi-

fied element.

4.1 Material models of early age concrete As the solidification theory, the properties of early age concrete are considered to vary proportionally to a volume function. The volume function models the in-creases in cement gel over time. The solidification theory (Bazant et al. 1989) was developed to investigate creep behavior in early age concrete. It is based on the com-pliance function, which is the deformation history for a constant stress. By using the same concept of solidified concrete at each time step, the solidified constitutive model can be proposed to determine the relation between stress and strain for analyzing the behavior of early age concrete structure before and after cracking.

A solidified constitutive model, as shown in Fig. 7, was constructed, where each element induces at different age and the height of each element expresses the mag-nitude of stress. The volume function, ( )tν , of concrete depends upon time and relates to cement hydration. The

Table 1 Properties of concrete and testing condition. Properties of concrete and testing conditions Value

Thermal conductivity, k (J/m⋅s⋅°C) 2.7 Specific heat, c (J/kg⋅°C) 1100 Thermal convection coefficient, h (J/m2⋅s⋅°C) 14 Density of concrete, ρ (kg/m3) 2300

Initial temperature (°C) 25 Atmosphere temperature (°C) 25 Poisson’s ratio 0.16 Thermal expansion coefficient (1/°C) 10×10-6

0 10 20 30

30

40

50

60

Time (days)

Tem

pera

ture

, (℃

) FEM #A FEM #B FEM #C RBSM #A RBSM #B RBSM #C

Fig. 5 Temperature distribution.

0 10 20 30-2

-1

0

1

Time, (days)

Hor

izon

tal s

tress

, (M

Pa)

FEM #A FEM #B FEM #C FEM #D RBSM #A RBSM #B RBSM #C RBSM #D

Fig. 6 Horizontal stress distribution.

σG

σG∆v1 ∆v2 ∆vn

Height of each element expresses magnitude of stress

Each element induces at different age

σG

σG∆v1 ∆v2 ∆vn

Height of each element expresses magnitude of stress

Each element induces at different age

Fig. 7 Concept of modified solidification.


solidified constitutive model is based on two assump-tions. The first assumption is that concrete elements solidified at different ages have the same normalized stress-strain relations no mater how young the elements are. This implies that stress-strain relations at each time change only the stress value depending on the volume function but strain remains constant. The other assump-tion is that, at each time step, the change in concrete strength depends on the incremental volume, d ( )itν , of concrete. Moreover, the material properties also depend on the volume function and have the same function as the volume function, such as Young’s modulus, com-pressive strength, tensile strength, and fracture energy.

The first assumption automatically leads to the con-clusion that the shape of solidified stress-strain relations are identical to the stress-strain relation at the virgin loading for completely hydrated concrete, which is de-noted as σ = f(ε). The second assumption leads to the following formulation,

( )τεεσσ vG ∆⋅∆⋅∆∆

=∑d

( ) ( )

( ) ( ) ετττεε

ε

τεεεε

τ

τ

dd

dd

0

0

⋅

⋅

∂∂⋅−

∂∂

=

⋅⋅−∂∂

=

∫

∫vf

vf

t

t

εd⋅= D (8)

where t = total time, τ = time at each step and (t)ν = specific total volume of solidified elements. σG and ε denote stress and strain of global load, respectively. This concept is simple and applicable for strain history de-pending on time before and after cracking.

Figure 8 shows the details and determination of local and global stress. As can be seen, local stress-strain re-lations are given by the global strain, ε(ti), corresponding to the time i and global stress, and σG(t) is given by the

superposition of all local stresses, σi(t), at the same time. As shown in Equation (8), the solidified constitutive model displays the elastic incremental method before cracking as well. Moreover, the difference between the solidified constitutive model and the elastic incremental method are not only the determination of the behavior after cracking but also the consideration of microcracks. Namely, the stresses calculated by the solidified consti-tutive model can consider the effect of microcracks. As shown in Fig. 8, σ1 and σ2 are in the softening branch of the solidified element. This means that microcracks take place. Therefore, the total stresses will be lower than the ones obtained from the elastic incremental method. Moreover, when total stress reaches the maximum value, cracking occurs. This is the criterion of cracking in con-crete of the solidified constitutive model.

4.2 Material models of hardened concrete The fracture criterion in RBSN is not based on a tensorial measure of stress, using instead the average stresses acting normally and tangentially to the particle interface. In the structural analysis, crack initiation is determined by the well known tensile strength criterion in which normal springs are set to represent the tensile and com-pressive behavior of concrete.

In this study, the tensile behavior of concrete up to the tensile strength is modeled by using a linear elastic, while a bilinear softening branch of a 1/4 model is assumed after cracking, as shown in Fig. 9(a), which is repre-sented by the tensile strength, ft, the tensile fracture en-ergy, Gft, and the distance between nuclei, h. The be-havior of concrete under compressive stress is modeled using a parabolic curve up to compressive strength f′c and a linear softening branch is assumed thereafter, as shown in Fig. 9(b). The slope of the linear softening branch is defined by considering the compressive fracture energy, Gfc, of Equation (9) (Nakamura and Higai 2001) to avoid mesh size dependence as well as tension behavior.

cfc fG ′= 8.8 (9)

Tangential springs represent the shear transfer mechanism of uncracked and cracked concrete. The shear strength is assumed to the Mohr-Coulomb type criterion with tension and compression caps, as shown in Fig. 10(a). The cohesion, c, and internal friction angle, φ, are set to 0.138f′c and 37°, respectively (Ueda et al. 1984). It is assumed that the shear strength is constant when normal stress is greater than ψf′c, where ψ set to 0.5. After shear stress reaches the yield strength, the stress moves on the yield surface until the shear strain reaches the ultimate strain, γu. The force in the shear spring is released and the local stiffness, kt, is set to zero when the shear strain exceeds the ultimate strain. The ultimate strain is set to 0.004µ in this study. The shear transfer capacity at crack interfaces depends on the crack opening (Saito et al. 1999), as shown in Fig. 10(b).

1t 2t 3t

t

t

1v2v3v0.1

( )1tε

( )2tε

( )tε

( )1tdv

( )3tdv( )2tdv

( ) cftdv ′1

( ) cftdv ′3

σv

2σ

( )tε

3σ1σ

( ) cftdv ′2

Stress-strain of each solidified element

Loading histories

Volume function

1t 2t 3t

t

t

1v2v3v0.1

( )1tε

( )2tε

( )tε

( )1tdv

( )3tdv( )2tdv

( ) cftdv ′1

( ) cftdv ′3

σv

2σ

( )tε

3σ1σ

( ) cftdv ′2

Stress-strain of each solidified element

Loading histories

Volume function

Fig. 8 Modified solidification concept.


5. Verification of solidified constitutive model

In this section, the solidified constitutive model was verified by setting the experiment and comparing its results with the numerical ones. To verify the concept and assumptions of the solidified constitutive model in the tensile behavior of concrete, three-point bend tests with notch beam specimens under various age and damage levels were carried out and the results were compared with the numerical ones obtained by using the unified numerical model under the same condition.

5.1 Test specimens and test procedures The notched beam specimens are 100 x 100 x 400 mm (width x depth x length) and had a notch extending over one-third of the specimen depth, as shown in Fig. 11. Their width is equal to 5 mm. The specimens are placed on two parallel roller supports and the load is applied uniformly along the thickness of the specimens. The crack mouth opening displacement (CMOD) and the applied load were recorded continuously during the test. A clip gauge was used to measure the CMOD. The loading rate was controlled through a constant CMOD increment rate.

For the first stage series of tests, the specimens were stored in a water bath at the desired temperature from the age of 24 hours after casting. The beam specimens were monotonically loaded until 2/3 of the maximum load within the post-peak region at 2 days and then unloaded. Thereafter, they were stored in a water bath again. These initial tests measure load versus CMOD under dis-placement control of the loading apparatus. The second stage series, which used the damaged specimens induced by the first monotonic loading, were conducted at 4 and 7 days until the specimens failed. Each series had 3 specimens. Bending tests without the unloading proce-dure (monotonic loading) were also carried out at 2, 4, and 7 days.

5.2 The volume function The variations in tensile strength and the fracture energy variation are shown in Fig. 12 according to the experi-mental results under monotonic loading. It was found that both fracture energy and tensile strength can be basically represented by a similar function. The volume function, Equation (10), was assumed to be the same as normalized tensile strength and fracture energy. There-fore, the second assumption of the solidified constitutive model is applicable for early age concrete.

1.175( )4.89

=+tv tt

(10)

5.3 The comparison results between experi-mental and numerical results The model of the three-point bend test using Rigid-Body-Spring Networks with 350 Voronoi poly-gons is shown in Fig. 13. For monotonic loading, beam specimens were analyzed at 2, 4, and 7 days and the comparison results of load-displacement (CMOD) rela-tions between experimental and numerical results are shown in Fig. 14. The shape of load-displacement (CMOD) and the maximum load corresponds well with the experimental results. For the initial-second loading, beam specimens were analyzed at 2 days for the initial loading, and at 4, 7 days for the second loading. The comparison results between the experimental and nu-merical results are shown in Fig. 15 and Fig. 16. The

εt1 εtu

σ

ft

εt0

ε

ft1

εt1 = 0.75Gft / fthεtu = 5.0Gft/ fthE

Gft/h

ft1 = 0.25ft

εε0/2 ε0 εcu

σ

Gfc/hEcEc

cf '

εt1 εtu

σ

ft

εt0

ε

ft1


Gft/h

ft1 = 0.25ft

εt1 εtu

σ

ft

εt0

ε

ft1


Gft/h

ft1 = 0.25ft

εε0/2 ε0 εcu

σ

Gfc/hEcEc

cf '

εε0/2 ε0 εcu

σ

Gfc/hEcEc

cf '

a) Tension behavior. b) Compressive behavior.Fig. 9 Stress-strain relations of concrete.

σ

τ

φ cψ ftcf ' cf '

τ

G

γ0 γu

τf

γ

ε = εtβ = 1.0

τ σσ τ

σ

τ

φ cψ ftcf ' cf ' σ

τ

φ cψ ftcf ' cf '

τ

G

γ0 γu

τf

γ

ε = εtβ = 1.0

τ σσ τ

τ

G

γ0 γu

τf

γ

ε = εtβ = 1.0

τ σσ τ

a) Mohr-Coulomb. b) Reduction of shear stiffness.Fig. 10 Shear model in concrete.

50 50300

100

Unit: mm

150 150

50 50300

100

Unit: mm

150 150

Fig. 11 Configuration of notched beam.

0 2 4 6 8 100

1

2

3

4

5

6

0

20

40

60

80

100

Tens

ile st

reng

th (M

Pa)

Concrete age (days)

Frac

ture

ene

rgy

(N/m

)

Tensile strength Fracture energy

Ft=6.51t/(4.89+t)|r|=0.979GF=123t/(4.89+t)|r|=0.977

Fig. 12 Tensile strength and fracture energy.


shape of load-displacement (CMOD) relations and the maximum load are also similar to the experimental re-sults for both the first and second loading curves. Therefore, the concept and the first assumption of the solidified constitutive model can be verified and can be applied for analyzing the problem of early age concrete structures to investigate the behaviors of such concrete structure before and after cracking depending on time and strain history.

6. Crack propagation analysis due to thermal stress

Wall concrete structures were modeled and analyzed by using the unified numerical model and the solidified constitutive model. Structural behaviors such as internal stress, deformations and crack patterns were compared with the experimental ones. The tests of real scale of wall concrete structures (specimens M1-M5) with 300 mm and 950 mm thicknesses, 15000 mm length and 2000 mm height were carried out by Ishikawa et al. (1989), as shown in Fig. 17. They investigated not only the con-struction joint behavior between newly cast concrete and hardened concrete, but also the effect of external restraint from hardened concrete by setting the different geome-tries of the wall structures.

It is a fact that the restraint variation depends on the cross-section areas of the structures, the geometric properties of the new concrete, possible slip failure of the construction joint, the modulus of elasticity of the newly and hardened concrete, and the translational and rola-tional boundary restraint situation from the foundation material (Nilson 2003). Therefore, to investigate the effect of slip failure at the construction joint, specimens M1 and M3 were selected for analyis in this paper be-cause they have the same cross-section areas and ge-ometry. The specimen M1 was constructed without ver-tical reinforcement in order to investigate the effect of external restraint from the hardened concrete with the construction joint, as shown in Fig. 17(b). The new concrete was placed on the top face of the hardened concrete using sand blasting. Specimen M3 was inves-

50 50300

150 150

100

P P(Displacement control)

Unit: mm50 50300

150 150

100

P P(Displacement control)

Unit: mm

Fig. 13 Numerical model.

0.2 0.4 0.6

1

2

3

4

5

6

0

2 days experimental results4 days experimental results7 days experimental results2 days numerical results4 days numerical results7 days numerical results

CMOD (mm)

Load

(KN

)

Fig. 14 Monotonic loading at 2, 4 and 7 days.

0.2 0.4 0.6

1

2

3

4

0

Load

(kN

)

CMOD (mm)

Experimental results Numerical results

Fig. 15 Load-CMOD relation of 2-4 days.

0.2 0.4 0.6

1

2

3

4

0

Load

(kN

)

CMOD (mm)

Experimental results Numerical results

Fig. 16 Load-CMOD relation of 2-7 days.

Hardened concreteNewly concrete

300

950Unit: mm.Slab

Adiabatic boundary

Hardened concreteNewly concrete

300

950Unit: mm.Slab

Adiabatic boundary a) Wall structure.

300

950 15000

1000

1000

M1

Sand blasting

New concrete

Hardened concrete

300

950 15000

1000

1000

M1

Sand blasting

New concrete

Hardened concrete

b) Specimen M1.

300

950 15000

1000

1000

Unit: mm

M3New concrete

Hardened concrete

300

950 15000

1000

1000

Unit: mm

M3New concrete

Hardened concrete

c) Specimen M3.

Fig. 17 Outline of experiment specimens.


tigated for the effect of external restraint by using verti-cal reinforcement between the new concrete and hard-ened concrete, as shown in Fig. 17(c). The reinforced bars were deformed bars with 19 mm diameter. The reinforcement ratios in the vertical and horizontal direc-tions were 0.767% and 0.573%, respectively. Table 2 lists the concrete properties of the new concrete of specimens M1 and M3 at 28 days.

For both specimens, Young’s modulus and Poisson’s ratio of hardened concrete were set to 20000 MPa and 0.17, respectively. Figure 18 shows the numerical model of specimens M1 and M3 with 1500 Voronoi polygons and the observation points A, B and C at the middle span. Points A, B, and C are 100 mm, 500 mm and 900 mm under the top surface of the new concrete, respectively. The bottom of the hardened concrete was assumed to be roller support and the tensile strength of the normal spring was set to 0.0315 MPa, taking into consideration the weight of the structures. For heat transfer analysis, the bottom surface was considered to behave as an adiabatic boundary and the other three boundaries as heat transfer boundaries. In addition, thermal insulating ma-terial was attached at the side surface of the specimen in experiment. For specimen M3, the reinforcing bar was not directly modeled in the numerical model. However, the tensile strength of the normal spring was given a sufficiently large value to avoid separation behavior at the construction joint. 6.1 Investigation of M3 specimen 6.1.1 Properties of concrete The volume function with respect to time can be derived from the Young’s modulus of the experimental results, as shown in Equation (11). It should be noted that the ma-terial properties were assumed depending on age, not on maturity and effective age.

( ) 25063 (4.577 3.799 )

=× +

tv tt

(11)

where t = time. For new concrete, the variations of ma-terial properties such as Young’s modulus, the tensile and compressive strength, and the fracture energy can be obtained by multiplying volume function and each property at 28 days based on the solidified constitutive model.

It should be noted that by neglecting creep or stress relaxation, thermal stresses may be overestimated by up to 70% (Slowik et al. 2005). Therefore, in this paper, the effect of creep at early ages was considered by using a correction factor of the Young’s modulus, ψ(t), proposed by the JSCE Standard, (JSCE. 2002). ψ(t) = 0.73 and 1.0 for up to 3 days and after 5 days, respectively. Parameter ψ was simply considered only in the compressive zone, as some researchers have reported that the tensile creep is small compared with the compressive creep (Goto et al. 1995).

6.1.2 Temperature distribution As mentioned before, in order to determine thermal stress in concrete structures, temperature distribution was determined first. A time-dependent adiabatic temperature rise was assumed in order to obtain the same temperature histories with experimental results as shown in Equation (12). The thermal concrete properties are listed in Table 3. The temperature distribution was calculated every 3 hours until 8 days.

( )teT 6.2139 −−×= (12)

where T = temperature and t = time. Figure 19 shows the comparison results of the temperature history of points A, B, and C. The coefficient of linear expansion of hardened concrete was set to 10.0×10-6/oC. According to the ex-

Table 2 Concrete properties of specimens M1 and M3 at 28 days.

Concrete properties M1 M3 Compressive strength, f′c (MPa) 32.4 30.1 Tensile strength, ft (MPa) 2.62 2.67 Young’s modulus, E (MPa) 25800 25500Tensile fracture energy, Gft (N/mm.)

0.1 0.1

A B C

Heat transfer boundaryAdiabatic boundary

A B C

Heat transfer boundaryAdiabatic boundary

Fig. 18 Numerical model of specimens M1 and M3.

Table 3 Thermal properties of concrete of specimen M3.

Thermal properties of concrete New concrete

Hardened concrete

Specific heat, c (J/kg⋅°C) 1700 850 Thermal conductivity, k (J/m⋅s⋅°C) 6.5 7.0 Thermal convection coefficient, h (J/m2⋅s⋅°C)

20.0 20.0

Density of concrete, ρ (kg/m3) 2400 2400 Initial temperature (°C) 30.0 30.0 Atmosphere temperature (°C) 26.0 26.0

0 2 4 6 8

30

40

50

60

Age (Days)

Tem

pera

ture

(℃）

Experimental results At point A At point B At point C

Numerical results At point A At point B At point C

Fig. 19 Temperature history of new concrete.


perimental results, the coefficient of linear expansion of new concrete depends on the time: 15.1×10-6/oC during the first 3 hours, 6.2×10-6/oC after 3 hours until 24 hours, and 8.9×10-6/oC after 24 hours. 6.1.3 Behavior of wall concrete structure From the experimental results, compressive stresses occurred before 1.5 days in the new concrete near the middle span and the specimen expanded. After that, stresses gradually changed to tensile stresses and the specimen gradually lifted up at both ends. The specimen shrank in the horizontal direction after the temperature passed the maximum temperature (at 1 day). At 4 days, a main crack occurred in the middle layer near the middle span of the new concrete. The main crack propagated upward and downward until through crack took place at 6 days. The strain gauges may have become damaged after the occurrence of through cracks, which may ex-plain why horizontal stresses did not suddenly drop, as shown in Fig. 22.

Figure 20 shows the deformations, crack patterns, and horizontal stress distributions from the numerical results. During the temperature increase (before 1 day), the middle span of the new concrete had compressive stresses and the specimen expanded. Moreover, the wall structure lifted up near both ends, which is similar to the numerical results obtained from Ishikawa et al. (1994) caused by the effect of the specimen length. When the temperature in the new concrete passed its peak (after 1 day), the expansion turned into a contraction and hori-zontal stresses in the new concrete gradually changed to tensile stresses. After 2 days, both ends of the wall were gradually lifted up and tensile stresses gradually in-creased. A main crack occurred near the middle span at 4 days. After that, the main crack continuously grew and propagated in the upward and downward direction at the same time, and the crack width gradually increased as well, as shown in Fig. 21. At 5.5 days, through cracks occurred and the crack width suddenly increased. The horizontal stresses adjacent to this crack abruptly de-creased. As a result, the position of the crack, the crack-ing dates, and the crack propagations were similar to the experimental ones, and the cracking behavior can be clearly seen step by step. These are the merit of the proposed model.

Figure 22 shows the comparison results of stress his-tories between the experimental and numerical results of points A, B and C. At point A, the stresses of the nu-merical results were higher than those of the experi-mental ones, since the difference of aging for each ele-ment due to the temperature gradient and history are not taken into consideration. However, the suddenly de-crease in horizontal stresses after the occurrence of through cracks can be clearly observed in the analysis at 5.5 days. Figure 23 displays the horizontal deformation of the new concrete and Figs. 24 and 25 show the vertical deformation at both ends of the new concrete. As a result, horizontal stress, horizontal deformations, and vertical

a) Deformations and crack patterns. b) Horizontal stress distribution.

Fig. 20 Behavior of wall structure.

2 4 6 8 10

0.1

0.2

0.3

0Age (Days)

Cra

ck w

idth

(mm

)

Upper layer Middle layer Lower layer

Fig. 21 Crack widths of through crack.

0 2 4 6 8-1

0

1

2

Experimental results Numerical results At point A At point A At point B At point B At point C At point C

Age (Days)

Hor

izon

tal s

tress

(MPa

)

Fig. 22 Horizontal stresses at middle.

0 2 4 6 8-2

-1

0

1

2Experimental results Numerical results

Upper Upper Middle Middle Lower Lower

Age (Days)

Hor

izon

tal d

efor

mat

ion

(mm

)

Fig. 23 Horizontal deformation histories.

0 2 4 6 8-1

0

1

Experimental results Upper Middle Lower

Numerical results Upper Middle Lower

Age (Days)

Ver

tical

def

orm

atio

n (m

m)

Fig. 24 Vertical deformation histories at left end.


deformations tend to be similar to the experimental re-sults. Ishikawa et al. (1997) analyzed the same structure using the FEM method with the creep function. As a result, horizontal stress distributions only before crack-ing were observed, similar to the results obtained from the unified numerical model. However, the proposed model is more simple than that model.

As mentioned before, the elastic incremental method has been normally used to analyze the behavior of early age concrete before cracking. Therefore, to clearly show the difference between the elastic incremental method and the solidified constitutive model, the specimen was calculated by using the elastic incremental method. The results are displayed in Fig. 26. As can be seen, hori-zontal stresses obtained from the elastic incremental method gave results that differed from the experimental ones because the elastic incremental method cannot consider the effect of microcracks. These points are the advantages of the solidified constitutive model. More-over, the sudden decrease of horizontal stresses after the occurrence of the through crack could not be observed as well.

6.2 Investigation of M1 specimen 6.2.1 Properties of concrete Similar to specimen M3, the volume function with re-spect to time of the M1 specimen can be derived from Young’s modulus obtained from the experiment, as shown in Equation (13).

( ) 25063 (3.371 3.853 )

=× +

tv tt

(13)

The reduction factor, ψ(t), of Young’s modulus con-sidering creep effect was also assumed to be the same as that of specimen M3.

6.2.2 Temperature distribution The time-dependent adiabatic temperature rise is as-sumed as shown in Equation (14). The thermal concrete properties are listed in Table 4.

( )teT 0.4140 −−×= (14)

The comparisons of temperature histories between the numerical and experimental results of the observation points are show in Fig. 27. The coefficient of linear ex-pansion of hardened concrete was set to 10.0×10-6/oC. According to the experimental results, the coefficient of linear expansion depends on time: 17.9×10-6/oC during the first 3 hours, 9.6×10-6/oC after 3 hours until 24 hours, and 9.8×10-6/oC after 24 hours. 6.2.3 Behavior of wall concrete structure With reference to the experimental results, compressive stresses occurred before 1.5 days in the new concrete near the middle span of the specimen and specimen ex-panded. After that, stresses gradually changed to tensile stresses. Cracks took place at both ends of the construc-tion joint at 2 days and both ends of the new concrete gradually lifted up. The specimen shrank in the hori-zontal direction after the temperature passed the maxi-mum temperature (at 1 day). A main crack occurred near the middle span of the new concrete at 3 days and through crack suddenly occurred. The main crack propagated in the upward and downward directions.

With the aim to simulate thermal cracking behavior of wall concrete structures with construction joints such as specimen M1, the modeling of construction joints relat-ing to normal and tangential springs is very important. Since it is difficult to define the mechanical properties of normal and tangential springs exactly, a parameter study

0 2 4 6 8-1

-0.5

0

0.5

1

Age (Days)

Ver

tical

def

orm

atio

n (m

m)

Experimental results Upper Middle Lower

Numerical results Upper Middle Lower

Fig. 25 Vertical deformation histories at right end.

0 2 4 6 8-1

0

1

2

3

Experimental results Elastic incremental At point A At point A At point B At point B At point C At point C

Age (Days)

Hor

izon

tal s

tress

(MPa

)

Fig. 26 Horizontal stresses at middle span of specimen.

Table 4 Thermal properties of concrete of specimen M1.

Thermal Properties of concrete New concrete

Hardened concrete

Specific heat, c (J/kg⋅°C) 1700 850 Thermal conductivity, k (J/m⋅s⋅°C) 5.0 2.7 Thermal convection coefficient, h (J/m2⋅s⋅°C)

4.5 2.0

Concrete density, ρ (kg/m3) 2400 2400 Initial temperature (°C) 30.0 30.0 Atmosphere temperature (°C) 26.0 26.0

0 1 2 3 4 5

30

40

50

60

Age (Days)

Tem

pera

ture

(℃）

Experimental results At point A At point B At point C

Numerical results At point A At point B At point C

Fig. 27 Temperature history of new concrete.


of each spring was performed and the effects of each mechanical property were investigated through com-parison with the experimental results.

Regarding to the properties of the normal spring, Ku-rihara et al. (1999) investigated the bond properties of hardened concrete joints under different surface condi-tions through tension softening diagrams in addition to conventional flexural bond strength. They concluded that the tensile fracture energy is a more sensitive index than the conventional flexural bond strength. As the experi-mental results obtained from hardened joints, the tensile strength and tensile fracture energy of normal springs at the construction joint were reduced to 80% and 75%, respectively. When these values were used for the con-struction joint between the hardened and new concrete to analyze wall concrete structures, cracks at the construc-tion joint did not occur. Furthermore, it was found that the higher the tensile fracture energy, the lower the separation behavior. Therefore, for the similarity to the experimental results of the wall concrete structure, the tensile strength and tensile fracture energy of the normal spring along the construction joint between the hardened and new concrete were set to the constants of 30% and 10% of that of new concrete, respectively.

On the other hand, since the behavior at the construc-tion joint not only depends on the normal spring but also the tangential spring, the properties of the tangential spring should be studied also by setting a different shear strength that is independent on the properties of the normal spring. In this study, shear strengths were cate-gorized into 3 cases, as shown in Fig. 28 by changing of tensile and compressive strength. In case 1, the tensile and compressive strengths were similar to normal spring. This means that the properties of the tangential spring depend on the normal spring. In case 2, both the tensile and compressive strengths were reduced to 30% and the shear strength automatically decreased, since compres-sive strength influences directly to the shear strength in the shear model. Therefore, the effect of shear strength can be investigated with this case. In case 3, only the tensile strength was reduced to 30% and the effect of the tensile strength can be evaluated when it is compared to Case 1.

By using the temperature histories obtained in section 6.2.2 with 3 cases of shear strength criteria, the defor-mations and crack patterns at 8 days of each case can be shown in Fig. 29. Moreover, Figs. 30, 31 and 32 show the horizontal stress distribution, horizontal deforma-tions, and vertical deformations for each case compared with the experimental results, respectively. It can be seen that the shear strength criteria influence the structural behaviors as follows. (1) Different structural behavior can be seen in Fig. 29 by comparing Case 2 and Case 3 with constant low tension cap (0.3ft). Case 2, which has low compression cap or low shear strength, shows slip behavior. There-fore, shear strength is very important for showing higher slip behavior. As can be seen in Fig. 31, the horizontal

deformations of Case 2 are similar to the experimental results due to the effect of slip behavior. According to the numerical results obtained from Ishikawa et al., (1994), when they set the stiffness of horizontal springs to zero, the new concrete of the wall structure did not lift up, which is similar with the numerical results in Case 2.

(2) By comparing Case 1 and Case 3 in Fig. 29, which have the same shear strength, clear separation behavior can be observed even though both cases have high and low tensile strengths. The vertical deformations of each

Case 3

Case 1

Case 2f`c ft

τ

Case 3

Case 1

Case 2f`c ft

τ

Fig. 28 Mohr-Coulomb of Cases 1-3.

Case 2: 0.3ft and 0.3f′c






Fig. 29 Deformations and crack patterns of each case at 8 days.

0 2 4 6 8

-1

0

1

Experimental Numerical Case 1 Numerical Case 2 At point A At point A At point A At point B At point B At point B At point C At point C At point C

Age (Days)

Hor

izon

tal s

tress

(MPa

)

0 2 4 6 8

-1

0

1

Experimental Numerical Case 3 At point A At point A At point B At point B At point C At point C

Age (Days)

Hor

izon

tal s

tress

(MPa

)

Fig. 30 Comparison of horizontal stress.


case are similar, as shown in Fig. 32. Therefore, the tensile strength slightly affects separation behavior at the construction joint.

(3) As can be seen in Fig. 30, the horizontal stress distributions of Case 1 and Case 3 have the same dis-tributions as the experimental results before through crack occurrence. Since through cracks did not occur, no sudden drop in the stresses could be observed. For Case 2, the horizontal stress distributions correspond well with the experimental results due to the effect of the slip behavior at the construction joint. As a result, horizontal stress slightly dropped at 2 days and suddenly dropped at 2.5 days due to slip behavior along the construction joint.

Figures 33 and 34 show the deformations, crack pat-terns, and horizontal stress distributions of Case 1 and

Case 2 obtained from the numerical results, respectively. For Case 1, as can be seen in Fig. 33, during the in-creasing temperature (before 1 day), the new and hard-ened concrete have compressive and tensile stress, re-spectively, and support inside the structure lifted up near both ends. At 1 day, cracks at the construction joint oc-curred at both ends. After the temperature passed the maximum temperature, the wall structure went into contraction and horizontal stresses in the new concrete gradually changed to tensile stresses. At the same time, cracks at the construction joint and separation behavior continuously occurred. At 3 days, many cracks took place at the new concrete as marked by a black circle resulting the non-uniform stress in the new concrete. The hardened concrete did not lift up because the external restraint was released by cracking at the construction joint. For Case 2, the behaviors of the deformations, crack patterns and horizontal stress distributions before 3 days were similar to the Case 1 specimen. But at 3 days, slip behavior occurred along the construction joint, re-sulting in the decrease of stresses along the construction joint as marked by the black circle in Fig. 34. The sudden drop in stress in Fig. 30 of the Case 2 specimen took place due to the effect of slip behavior instead of through crack behavior.

7. Conclusion

(1) A time-dependent structural analysis method and constitutive model to determine thermal cracking be-havior and thermal stress before and after cracking were

0 2 4 6 8-2

-1

0

1

2Experimental results

Upper Middle Lower

Numerical results of Case 1 Upper Middle Lower

Age (Days)

Hor

izon

tal d

efor

mat

ion

(mm

)


0 2 4 6 8-2

-1

0

1


Upper Middle Lower


Age (Days)

Hor

izon

tal d

efor

mat

ion

(mm

)

0 2 4 6 8-2

-1

0

1


Upper Middle Lower


Age (Days)

Hor

izon

tal d

efor

mat

ion

(mm

)


0 2 4 6 8-2

-1

0

1


Upper Middle Lower


Age (Days)

Hor

izon

tal d

efor

mat

ion

(mm

)

Fig. 31 Comparison of horizontal deformations.

0 2 4 6 8

0

0.5

1

Experimental results Numerical results of Case 1 Upper Upper Middle Middle Lower Lower

Age (Days)

Ver

tical

def

orm

atio

n (m

m)


0 2 4 6 8

0

0.5

1


Age (Days)

Ver

tical

def

orm

atio

n (m

m)

0 2 4 6 8

0

0.5

1


Age (Days)

Ver

tical

def

orm

atio

n (m

m)


0 2 4 6 8

0

0.5

1


Age (Days)

Ver

tical

def

orm

atio

n (m

m)

Fig. 32 Comparison of vertical deformations at section A.


Fig. 33 Behavior of wall structure in Case 1.


Fig. 34 Behavior of wall structure in Case 2.


developed. The time-dependent structural model consists of a solidified constitutive model for early age concrete and a unified numerical model. To simulate cracking behaviors of early age concrete, the fundamental con-stitutive model was based on a solidification concept that is different from the one presented by Bazant et al. (1989) in that the solidified constitutive model can con-sider the nonlinear stress-strain relation easily and stress increments in each solidified element can be calculated. Moreover, to evaluate cracks in early concrete structures, the unified numerical model (Nakamura et al. 2006) consisting of the Rigid-Body-Spring Networks (RBSN) model and the truss network model was used with Vo-ronoi random mesh.

(2) To verify the unified numerical model and the solidified constitutive model, two-dimensional concrete specimen and three-point bend test with notched beam were constructed, respectively. The results of the two-dimensional concrete specimen gave good results corresponding to the FEM method. Moreover, the nu-merical results of the notched beam specimens using RBSM with the solidified constitutive model also showed good agreement with the experimental results. Therefore, it can be concluded that the unified numerical model can be used to evaluate structural behavior considering heat transfer problem and the solidified constitutive model can be also applied to analyze the problem of early age con-crete with strain history to investigate the behavior of concrete structures before and after concrete cracking. (3) The cracking behaviors of wall structures obtained from the numerical results reasonably correspond with the experimental behavior. For specimen M3, the numerical results show bending behavior at both ends of the specimen, as well as cracking and crack propagation that are similar to the experimental ones. Therefore, the uni-fied numerical model combined with the solidified con-stitutive model can be used to analyze thermal stress in mass concrete as well as thermal crack propagation. For specimen M1, separation and slip behavior were inves-tigated using parameter study. As a result, the lower the fracture energy, the higher the separation behavior. Moreover, the lower the shear strength of Mohr-Coulomb, the higher the slip behavior. (4) From parameter analysis in section 6.2, it was found that wall structural behaviors with construction joint depend on interface characteristics such as shear and tensile strength. Therefore, testing of the interface char-acteristics between hardened and early age concrete should be performed. (5) As the proposed model is in the first stage of de-velopment, the prime objectives of this paper were to propose and verify the possibility of the new numerical model. Realizing the importance of creep and shrinkage, the authors will conduct further studies to consider these issues in greater detail for more accurate prediction of thermal stress and cracking behaviors.

Acknowledgments Sincere gratitude is expressed to Prof. Ishikawa Masami for the experimental results of wall concrete structures and his cooperation. The authors are also grateful to Electro Technology of Chubu for supporting this re-search. References Bazant, Z. P. and Prasannan, S. (1989). “Solidification

theory for concrete creep I: Formulation.” Journal of Engineering Mechanics, ASCE, 115(8), 1691-1703.

Bolander, J. E. and Berton, S. (2004). “Simulation of shrinkage induced cracking in cement composite overlays.” Cement and Concrete Composites, 26(7), 861-871.

Bolander, J. E. and Saito, S. (1998). “Fracture analysis using spring networks with random geometry.” Engineering Fracture Mechanics, 61, 569-591.

Cusson, D. (2001). “Sensitivity analysis of the early-Age properties of high- performance concrete - A case study on bridge barrier walls.” 6th International Conference on Creep, Shrinkage & Durability Mechanics of Concrete and Other Quasi-Brittle Materials, Cambridge MA, 20-22 August 2001.

Goto, T., Uehara, T. and Umehara, H. (1995). “Studied on creep behavior of early age concrete.” Proceedings of the Japan Concrete Institute, 17(1), 1133-1138.

Ishikawa, M., Maeda, T., Nishioka, T. and Tanabe, T. (1989). “An experimental study on thermal stress and thermal deformation of massive concrete.” Proceedings of JSCE, 11(408), 121-130.

Ishikawa, M. and Tanabe, T. (1994). “Study of external restraint of mass concrete.” In Thermal Cracking In Concrete at Early Age, Proceedings of the International RILEM Symposium, Ed. by R. Springenschmid, Munich, 187-194.

Ishikawa, Y., Kikukawa, H., Ishikawa, M. and Tanabe, T. (1997). “Development of plasticity model for interface characteristics and thermal stress analysis of massive layer concrete structure.” Transactions of the Japan Concrete Institute, 19, 65-72.

JCI, (2005). “Japan Concrete Institute Computer Program of Thermal Stress Analysis for Massive Concrete Structure (JCMAC1).” Tokyo: Japan Concrete Institute.

Kawai, T. (1978). “New discrete models and their application to seismic response analysis of structures.” Nuclear Engineering and Design, Englewood Cliffs, New Jersey: Prentice-Hall; 48, 207-229.

Kurihara, N., Kunieda, M., Uchida, Y. and Rokugo, K. (1999). “Bond properties of concrete joints and size effect.” Journal Material, Concrete Structure, Pavements, JSCE, 42(613), 309-318.

Mihashi, H. and Leite, J. P. de B. (2004). “State-of-the-art report on control of cracking in early age concrete.” Journal of Advanced Concrete Technology, 2(2), 141-145.


Nakamura, H. and Higai, T. (2001). “Compressive fracture energy and fracture zone length of concrete.” In: Modeling of inelastic behavior of RC structures under seismic loads. ASCE, 471-487.

Nakamura, H., Srisoros, W., Yashiro, R. and Kunieda, M. (2006). “Time-dependent structural analysis considering mass transfer to evaluate deterioration process of RC structures.” Journal of Advanced Concrete Technology, 4(1), 147-158.

Ngo, D. and Scordelis, A. C. (1967). “Finite element analysis of reinforced concrete beams.” American Concrete Institute Journal, 64(3), 152-163.

Nilson, M. (2003). “Restraint factors and partial coefficients for crack risk analyses of early age concrete structures.” Technical Report, Department of Civil and Mining Engineering, Division of Structural Engineering, Lulea University of Technology.

Noda, T. (2004). “Lattice equivalent continuum model

considering the initial stress of RC structures.” Proceedings of the Japan Concrete Institute, 26(2), 1129-1134. (in Japanese)

Saito, S. (1999). “Fracture Analyses of Structural Concrete Using Spring Network with Random Geometry.” Doctoral thesis, Kyushu University.

Slowik, V., Schmidt, D. and Nietner, L. (2005). “Effect of stress relaxation on thermal stresses.” The Seventh International Conference on Creep, Shrinkage and Durability of Concrete and Concrete Structure (CONCREEP7), 411-416.

Ueda, M., Takeuchi, H., Higuchi, H. and Kawai, T. (1984). “A discrete limit analysis of reinforced concrete structures.” In Computer-Aided Analysis and Design of Concrete Structure, eds. Damjanic, F., Hinton, E., Owen, D. R. J., Bicanic, N. and Simovic, V., Pineridge Press, Swansea, UK, 1369-1384.

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